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Title: Functions, Limits and Continuity Prof. Dr. Nasima Akhter And Md. Masum Murshed Lecturer Department of Mathematics, R.U. 29 July, 2011, Friday Time: 6:00 pm-7:30 pm 1

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Outline Functions and its graphs. One-one, Onto and inverse functions. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 2

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Outline Functions and its graphs. One-one, Onto and inverse functions. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 3

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Functions and its Graphs 4 X Y f x y = f (x)

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Functions and its Graphs 5 f(x) x

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Functions and its Graphs 6 X Y f x y = f (x) f : X → Y if for each x ∊ X ∃ a unique y ∊ Y such that y = f(x).

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Functions and its Graphs 7 f f f f is a function f f is not a function f is a function f

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Functions and its Graphs 8 X Y f x y = f (x) f : X → Y if for each x ∊ X ∃ a unique y ∊ Y such that y = f(x). Domain of f Range of f Co-domain of f f[X] = {f(x) : x ∊ X} = R Real valued function

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Functions and its Graphs 9 X Y f y = f (x) f : X → Y if for each x ∊ X ∃ a unique y ∊ Y such that y = f(x). Domain of f Range of f Co-domain of f f[X] = {f(x) : x ∊ X} Set function Class of sets

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Functions and its Graphs 10 X Y f y = f (x) If f : X → Y then ∃ two set functions f : 2 X → 2 Y and f -1 : 2 Y → 2 X Domain of f Range of f Co-domain of f f[A] = {f(x) : x ∊ A } A A f(A)

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Functions and its Graphs 11 X Y f B B f -1 (B) f -1 (B) = {x ∊ X : f(x) ∊ B}. If f : X → Y then ∃ two set functions f : 2 X → 2 Y and f -1 : 2 Y → 2 X

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Functions and its Graphs 12 f f ({1, 3, 4}) = {a, b, d}, f ({1, 2}) = {a}, also f ({2, 3}) = {a, d} f -1 ({a})= {1, 2}, f -1 ({a, b, d})= {1, 2, 3, 4}, f -1 ({b, c})= {4}, f -1 ({c})= Ø,

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Functions and its Graphs 13 Between these two set functions f -1 plays very important role in topology and measure. Since f -1 preserves countable union, countable intersection, difference, monotonicity, complementation etc. i.e. Theorem If f : X → Y then for any subset A and B of Y, (i) f -1 (A ∪ B) = f -1 (A) ∪ f -1 (B). (ii) f -1 (A ∩ B) = f -1 (A) ∩ f -1 (B). (iii) f -1 (A ∖ B) = f -1 (A) ∖ f -1 (B). (iv) If A ⊆ B then f -1 (A) ⊆ f -1 (B). (v) f -1 (A c ) = (f -1 (A)) c. And, more generally, for any indexed {A i } of subsets of Y, (vi) f -1 ( ∪ i A i ) = ∪ i f -1 (A i ). (vii) f -1 ( ∩ i A i ) = ∩ i f -1 (A i ).

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Functions and its Graphs 14 x x y y X Y X ⨉ Y = {(x, y) : x ∊ X, y ∊ Y } (x, y) A relation from X to Y is a subset of X ⨉ Y.

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Functions and its Graphs 15 A function can also be described as a set of ordered pairs (x, y) such that for any x- value in the set, there is only one y-value. This means that there cannot be any repeated x-values with different y-values. A relation is called a function if for any x-value in the set, there is only one y-value. This means that there cannot be any repeated x-values with different y-values. f f is not a function g g is a function f = {(1, a), (1, b), (2, c), (3, c), (4, d)} is not a function. g = {(1, a), (2, a), (3, d), (4, b)} is a function.

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Functions and its Graphs 16 y = f(x) X, A subset of R We plot the domain X on the x-axis, and the co-domain Y on the y-axis. Then for each point x in X we plot the point (x, y), where y = f(x). The totality of such points (x, y) is called the graph of the function. (x, y) x x f Y, A subset of R

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Functions and its Graphs 17 This is the graph of a function. All possible vertical lines will cut this graph only once. This is not the graph of a function. The vertical line we have drawn cuts the graph twice. The Vertical Line Test

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Functions and its Graphs Now we consider some examples of real functions. Example 1. The identity function. Let the function f : R → R be defined by f(x) = x for all real x. This function is often called the identity function on R and it is denoted by 1 R. Its domain is the real line, that is, the set of all real numbers. Here x = y for each point (x, y) on the graph of f. The graph is a straight line making equal angles with the coordinates axes (see Figure-1 ). The range of f is the set of all real numbers. Figure-1 Graph of the identity function f(x) = x. 18

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Functions and its Graphs Example 2. The absolute-value function. Consider the function which assigns to each real number x the nonnegative number |x|. We define the function y = |x| as From this definition we can graph the function by taking each part separately. The graph of y = |x| is given below. Figure- 2 Graph of the absolute-value function y = f(x) = |x|. 19

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Functions and its Graphs Example 3. Constant functions. A function whose range consists of a single number is called a constant function. An example is shown in Figure-3, where f(x) = 3 for every real x. The graph is a horizontal line cutting the Y-axis at the point (0, 3). Figure- 3 Graph of the constant function f(x) = 3. 20

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Functions and its Graphs Example 4. Linear functions and affine linear function. Let the function g be defined for all real x by a formula of the form g(x) = ax + b, where a and b are real numbers, then g is called a linear function if b = 0. Otherwise, g is called a affine linear function. The example, f(x) = x, shown in Figure-1 is a linear function. And, g(x) = 2x - 1, shown in Figure-4 is a affine linear function. Figure- 4 Graph of the affine linear function g(x) = 2x - 1. 21

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Functions and its Graphs 22 Example 5. The greatest integer function The greatest integer function is defined by f(x) = [x] = The greatest integer less than or equal to x. Figure- 5 shows the graph of f(x) = [x]. Figure- 5 Graph of the greatest integer function is defined by f(x) = [x].

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Functions and its Graphs Example 6. Polynomial functions. A polynomial function P is one defined for all real a by an equation of the form P(x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + ……………+ c n x n The numbers c 0, c 1, c 2, c 3,……………,c n are called the coefficients of the polynomial, and the nonnegative integer n is called its degree (if c n ≠ 0). They include the constant functions and the power functions as special cases. Polynomials of degree 2, 3, and 4 are called quadratic, cubic, and quartic polynomials, respectively. Figure-6 shows a portion of the graph of a quartic polynomial P given by P(x) = ½ x 4 – 2x 2. Figure- 6 Graph of a quartic polynomial function p(x) = ½ x 4 – 2x 2. 23

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Functions and its Graphs 24 fg are functions h 1 = f +g h 2 = f.g h 3 = f /g h 7 = g ∘ f h 6 = f ∘ g k = constant h 4 = k + f h 5 = k. f

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Functions and its Graphs 25 f and g are real valued and defined on the same domain X and k is a real number. if h 1 = f +g,h 2 = f.g,h 3 = f /g, h 7 = g ∘ f h 6 = f ∘ g and We can define h 1 (x)= (f +g)(x) = f(x) + g(x), by h 2 (x)= (f.g)(x) = f(x). g(x), h 3 (x)= (f /g)(x) = f(x) / g(x), Also we can define by h 6 (x)= (f ∘ g)(x) = f(g(x)) if co-domain of g = domain of f and h 7 (x)= (g ∘ f)(x) = g (f(x)) if co-domain of f = domain of g Generally, f ∘ g ≠ g ∘ f h 4 = k + f andh 5 = k. f h 4 (x)= (k + f)(x) = k + f(x), h 5 (x)= (k. f)(x) = k. f(x),

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Composition Function 26 If f : X → Y and g : Y → Z then we define a function (g ∘ f) : X → Z by (g ∘ f)(x) ≡ g(f(x)).

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Composition Function Example Let f : X → Y and g : Y → Z be define by the following diagrams Then we can compute (g ∘ f) : X → Z by its definition: (g ∘ f)(a) ≡ g(f(a)) = g(y) = t (g ∘ f)(b) ≡ g(f(b)) = g(z) = r (g ∘ f)(c) ≡ g(f(c)) = g(y) = t Remark: Let f : X → Y. Then 1 Y ∘ f = f and f ∘ 1 X = f where 1 X and 1 Y are identity functions on X and Y respectively. That is the product of any function and the identity function is the function itself. 27

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Functions and its Graphs 28 Theorem Two functions f and g are equal if and only if (a) f and g have the same domain, and (b) f(x) = g(x) for every x in the domain of f. Theorem: Let f : X → Y, g : Y → Z and h : Z → W. Then (h ∘ g) ∘ f = h ∘ (g ∘ f).

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Outline Functions and its graphs. One-one, Onto and inverse functions. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Semi-continuity, Lipschitz continuity. 29

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One-one Function 30 Different elements of X Different elements of Y X Y f is one-one f : X → Y is said to be one-one if f(x) = f(y) implies x = y or, equivalently, x ≠ y implies f(x) ≠ f(y).

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One-one Function Examples (vi) Let X = {1, 2, 3, 4} and Y = {a, b, c, d, e} and f and g be two functions from X into Y given by the following diagrams 31 Here f is not a one-one function since a is the image of two different elements 1 and 2 of X. Here g is a one-one function since different elements of X have different images.

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Functions and its Graphs 32 This is the graph of a one- one function. All possible horizontal lines will cut this graph only once. This is not the graph of a one- one function. The horizontal line we have drawn cuts the graph twice. The Horizontal Line Test

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One-one Function 33 Examples The function f : R → R defined by f(x) = x 2 is not a one-one function since f(2) = f(-2) = 4. Figure- 7 Graph of the function f(x) = x 2.

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One-one Function 34 Examples The function f : R → R defined by f(x) = e x is a one-one function. Figure- 8 Graph of the function f(x) = e x. Proof: Let f(x) = f(y) then e x = e y i.e. x = y Hence by definition f is a one-one function.

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One-one Function 35 Examples The absolute-value function f : R → R defined by f(x) =|x |is not a one-one function since f(2) = f(-2) = 2. Figure- 9 Graph of the absolute-value function y = f(x) = |x|.

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One-one Function 36 Examples The identity function f : R → R defined by f(x) = x is a one-one function. Figure- 10 Graph of the identity function f(x) = x. Proof: Let f(x) = f(y) then x = y Hence by definition f is a one-one function.

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Onto Function 37 Every element of Y is the image of some element of X X Y f is onto f : X → Y is said to be onto if for every y ϵ Y, ∃ an element x ϵ X such that y = f(x), i.e. f(X) = Y.

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Onto Function Examples (vi) Let X = A = {1, 2, 3, 4}, Y = {a, b, c} and B= {a, b, c, d, e} and f and g be two functions from X into Y and A into B respectively given by the following diagrams 38 Here f is a onto function since every element of Y appears in the range of f Here g is not a onto function since e ∊ B is not an image of any element of A.

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Onto Function 39 This is the graph of a onto function. All possible horizontal lines will cut this graph at least once. This is not the graph of a onto function. the horizontal line drawn above does not cut the graph. The Horizontal Line Test

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Onto Function 40 Examples The function f : R → R defined by f(x) = e x is a not an onto function. Figure- 11 Graph of the function f(x) = e x. Proof: Since -2 is an element of the co-domain R then there does not exist any element x in the domain R such that -2 = e x = f(x). Hence by definition f is not a onto function.

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Onto Function 41 Examples The identity function f : R → R defined by f(x) = x is a onto function. Figure- 12 Graph of the identity function f(x) = x. Proof: Since for every y in the co-domain R, ∃ an element x in the domain R such that y = f(x). Hence by definition f is a onto function.

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Onto Function 42 Examples The function f : R → R defined by f(x) = x 2 is not an onto function. Figure- 13 Graph of the function f(x) = x 2. Proof: Since -2 is an element of the co-domain R then there does not exist any element x in the domain R such that -2 = x 2 = f(x). Hence by definition f is not a onto function.

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Inverse Function 43 X Y f is one-one and onto x y = f(x) f f -1

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Inverse Function 44 YX Y X Example Let f : X → Y be define by the following diagram Here f is one-one and onto. Therefore f -1, the inverse function, exists. We describe f -1 : Y → X by the diagram

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Inverse Function Theorem on the inverse function. Theorem 1 Let the function f : X → Y be one-one and onto; i.e. the inverse function f -1 : Y → X exists. Then the product function (f -1 ∘ f ) : X → X is the identity function on X, and the product function (f ∘ f -1 ) : Y → Y is the identity function on Y, i.e. (f -1 ∘ f ) = 1 X and (f ∘ f -1 ) = 1 Y. Theorem 2 Let f : X → Y and g : Y → X. Then g is the inverse of f, i.e. g = f -1, if the product function (g ∘ f ) : X → X is the identity function on X, and the product function (f ∘ g) : Y → Y is the identity function on Y, i.e. (g ∘ f ) = 1 X and (f ∘ g ) = 1 Y. 45

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Outline Functions and its graphs. One-one, Onto and inverse functions. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 46

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Algebraic and Transcendental Functions 47 Algebraic functions Algebraic functions are functions y = f (x) satisfying an equation of the form p 0 (x)y n + p 1 (x)y n-1 +... + p n-1 (x)y + p n (x) = 0 …………………………… (1) where p 0 (x),..., p n (x) are polynomials in x. If the function can be expressed as the quotient of two polynomials, i.e., P(x)/Q(x) where P(x) and Q(x) are polynomials, it is called a rational algebraic function; otherwise, it is an irrational algebraic function. Transcendental functions Transcendental functions are functions which are not algebraic; i.e., they do not satisfy equations of the form of Equation (1). Example is an algebraic function since it satisfies the equation (x 2 – 2x + 1)y 2 + (2x 2 – 2x)y + (x 2 – x) = 0.

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Algebraic and Transcendental Functions 48 The following are sometimes called elementary transcendental functions. 1. Exponential function: f (x) = a x, a ≠ 0, 1. 2. Logarithmic function: f (x) = log a x, a ≠ 0, 1. This and the exponential function are inverse functions. If a = e = 2.71828..., called the natural base of logarithms, we write f (x) = log e x = In x, called the natural logarithm of x. 3. Trigonometric functions: The variable x is generally expressed in radians (π radians = 180 ∘ ). For real values of x, sin x and cos x lie between –1 and 1 inclusive. 4. Inverse trigonometric functions. The following is a list of the inverse trigonometric functions and their principal values:

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Outline Functions and its graphs. One-one, Onto and inverse functions. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 49

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Bounded Function Bounded function A function f defined on some set X with real values is called bounded, if the set of its values is bounded. In other words, there exists a real number M < ∞ such that |f(x)| ≤ M or –M ≤ f(x) ≤ M for all x in X. Geometrically, the graph of such a function lies between the graphs of two constant step functions s and t having the values — M and +M, respectively. Figure- 14 Graph of a bounded function. 50 Intuitively, the graph of a bounded function stays within a horizontal band.

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Bounded Function Figure- 16 Graph of a bounded below function. 51 Bounded above function and Bounded below function Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x) ≥ B for all x in X, then the function is said to be bounded below by B. Figure- 15 Graph of a bounded above function.

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Bounded Function 52 Unbounded function The function which is not bounded is called unbounded function. Figure- 17 Graphs of un-bounded functions.

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Monotonic Function Monotonic functions A function f is said to be increasing on a set S if f(x) ≤ f(y) for every pair of points x and y in S with x ≤ y. If the strict inequality f(x) < f(y) holds for all x < y in S, the function is said to be strictly increasing on S. Similarly, f is called decreasing on S if f(x) ≥ f(y) for all x ≤ y in S. If f(x) > f(y) for all x < y in S, then f is called strictly decreasing on S. A function is called monotonic on S if it is increasing on S or if it is decreasing on S. The term strictly monotonic means that is strictly increasing on S or strictly decreasing on S. Ordinarily, the set S under consideration is either an open interval or a closed interval. Figure- 18 Monotonic functions. 53

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Monotonic Function Piecewise Monotonic functions A function f is said to be piecewise monotonic on an interval if its graph consists of a finite number of monotonic pieces. That is to say, f is piecewise monotonic on [a, b] if there is a partition P of [a, b] such that f is monotonic on each of the open subintervals of P. In particular, step functions are piecewise monotonic, as are all the examples shown in Figures- 23 and 24. Figure- 19 Piecewise Monotonic function. 54

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Monotonic Function Example 1. The power functions. If p is a positive integer, we have the inequality x p < y p if 0 ≤ x < y, which is easily proved by mathematical induction. This shows that the power function f, defined for all real x by the equation f(x) = x p, is strictly increasing on the nonnegative real axis. It is also strictly monotonic on the negative real axis (it is decreasing if p is even and increasing if p is odd). Therefore, f is piecewise monotonic on every finite interval. Example 2. The square-root function. Let f(x) = √x for x > 0. This function is strictly increasing on the nonnegative real axis. In fact, if 0 ≤ x < y, we have Example 3. The graph of the function g defined by the equation g(x) = √(r 2 -x 2 ) if — r ≤ x ≤ r is a semicircle of radius r. This function is strictly increasing on the interval — r ≤ x ≤ 0 and strictly decreasing on the interval 0 ≤ x ≤ r. Hence, g is piecewise monotonic on [-r, r]. 55

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Outline Functions and its graphs. One-one, Onto and inverse functions. Principal values. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 56

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Limit of a Function 57 Example

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Limit of a Function 58 Example

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Limit of a Function 59 Figure- 20

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Limit of a Function 60 Example Figure- 21

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Limit of a Function 61 Example Figure- 22

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Limit of a Function 62 Example Figure- 23

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Limit of a Function 63 Neighborhood of a point Any open interval containing a point p as its midpoint is called a neighborhood of p. The neighborhood of p with radius r is denoted by N r (p) and defined by N r (p) = {x ϵ R : |x - p| < r} = (p – r, p + r). Deleted neighborhood of a point N r *(p) = N r (p) \ {p} is called the deleted neighborhood of p, i.e. N r *(p) = {x ϵ R : x ≠ p and |x - p| < r}.

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Limit of a Function 64 Or equivalently if for every neighborhood N ε (L) there is some deleted neighborhood N δ (a) such that if x ϵ N δ (a) then f(x) ϵ N ε (L), i.e. for every neighborhood N ε (L) there is some neighborhood N δ (a) such that if x ϵ N δ (a) \ {a} then f(x) ϵ N ε (L). Or equivalently if for every neighborhood N ε (L) there is some deleted neighborhood N δ (a) such that if x ϵ N δ (a) then f(x) ϵ N ε (L), i.e. for every neighborhood N ε (L) there is some neighborhood N δ (a) such that if x ϵ N δ (a) \ {a} then f(x) ϵ N ε (L). Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever 0 < |x-a| < δ Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever 0 < |x-a| < δ

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Limit of a Function 65 Figure- 24 f(x) L L - ε L + ε a a - δa + δ

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Limit of a Function Find the value of delta ( ) and Epsilon ( ) 66 Figure- 25

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Limit of a Function 67 Procedure for finding limit Step-1 Guess L Step-2 Choose an arbitrary (small) number ε > 0. Step-3 Find the value of δ > 0 which satisfies the following condition |f(x) - L|< ε whenever 0 < |x-a| < δ

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Limit of a Function Example Use the definition of the limit to prove the following limit. Solution: Let ε > 0 be any arbitrary (small) number then we need to find a number δ > 0 so that the following will be true. |(5x - 4) - 6|< ε whenever 0 < |x-2| < δ Simplifying the left inequality in an attempt to get a guess for δ, we get, |(5x - 4) - 6| = |5x - 10| = 5|x - 2| < ε ⇒ |x - 2| < ε/5. So, it looks like if we choose δ = ε/5 we should get what we want. Let’s now verify this guess. So, again let ε > 0 be any number and then choose δ = ε/5. Now, assume 0 < |x-2| < δ = ε/5 then we get, |(5x - 4) - 6| = |5x - 10| = 5|x - 2| < 5(ε/5) = ε. So, we’ve shown that |(5x - 4) - 6|< ε whenever 0 < |x-2| < δ = ε/5 Hence by definition, 68

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Limit of a Function 69

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Limit of a Function 70 Example Use the definition of the limit to prove the following limit. Solution: Let ε > 0 be any arbitrary (small) number then we need to find a number δ > 0 so that the following will be true. | - 4| < ε whenever 0 < |x-2| < δ Since x ≠ 2 then for all, f(x) = Assume δ ≤ 1, then |x-2| < δ implies 1 < x < 3. Thus So, we can choose δ = ε. Then we have, |f(x) - 4|< ε whenever 0 < |x-2| < δ Hence by definition,

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Limit of a Function 71 Figure- 26 4 - ε 4 + ε 2 - δ 2 + δ

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Limit of a Function 72 Solution: Let ε > 0 be any arbitrary (small) number then we need to find a number δ > 0 so that the following will be true. |f(x) - 0|< ε whenever 0 < |x-0| < δ Simplifying the left inequality in an attempt to get a guess for δ, we get, =since1 So, we can choose δ = ε. Then we have, |f(x) - 0|< ε whenever 0 < |x-0| < δ Hence by definition,

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Limit of a Function 73 Theorem Proof

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Outline Functions and its graphs. One-one, Onto and inverse functions. Principal values. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 74

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Right hand limit 75 Or equivalently if for every neighborhood N ε (L) there is some number δ > 0 such that if x ϵ (a, a + δ) then f(x) ϵ N ε (L). Or equivalently if for every neighborhood N ε (L) there is some number δ > 0 such that if x ϵ (a, a + δ) then f(x) ϵ N ε (L). Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then for right hand limit we say, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever 0 < x-a < δ (Or a < x < a + δ) Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then for right hand limit we say, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever 0 < x-a < δ (Or a < x < a + δ)

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Right hand limit 76 Figure- 27 f(x) L L - ε L + ε a a + δ

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Right hand limit Example Use the definition of the limit to prove the following limit. Solution Let ε > 0 be any arbitrary (small) number then we need to find a number δ > 0 so that the following will be true. | √ x - 0|< ε whenever 0 < x-0 < δ i.e. we need to show that | √ x|< ε whenever 0 < x < δ Simplifying the left inequality in an attempt to get a guess for δ, we get, | √ x|< ε ⇒ x < ε 2. So, it looks like we can choose δ = ε 2. Let’s now verify this guess. So, again let ε > 0 be any number and then choose δ = ε 2. Now, assume 0 < x- 0 < δ = ε 2 then we get, | √ x - 0| = | √ x| < | √ ε 2 | = ε. So, we’ve shown that | √ x - 0|< ε whenever 0 < x-0 < δ = ε 2 Hence by definition, 77

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Left hand limit 78 Or equivalently if for every neighborhood N ε (L) there is some number δ > 0 such that if x ϵ (a - δ, a ) then f(x) ϵ N ε (L). Or equivalently if for every neighborhood N ε (L) there is some number δ > 0 such that if x ϵ (a - δ, a ) then f(x) ϵ N ε (L). Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then for left hand limit we say, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever – δ < x-a < 0 (Or a - δ < x < a) Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then for left hand limit we say, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever – δ < x-a < 0 (Or a - δ < x < a)

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Left hand limit 79 Figure- 28 f(x) L L - ε L + ε a a - δ

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Right and left hand limits 80 Figure- 29 4 - ε 4 + ε 2 - δ 2 + δ =

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Right and left hand limits 81 Example Figure- 30 x f(x) 1

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Right and left hand limits. 82 That is the definition of left hand limit is satisfied for δ = ε.

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Right and left hand limits 83 Figure- 31 Figure- 32

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Right and left hand limits 84 L + - Figure- 33

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Right and left hand limits 85 f(x) Figure- 34

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Right and left hand limits 86

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Limit Properties 87

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Limit Properties 88 Using the limit properties We can easily find the following limits

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Limit Properties 89 We can take this fact one step farther to get the following theorem.

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Limit Properties 90 Figure- 35 The following figure illustrates what is happening in Squeeze Theorem. The Squeeze theorem is also known as the Sandwich Theorem and the Pinching Theorem.

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Outline Functions and its graphs. One-one, Onto and inverse functions. Principal values. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 91

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Special Limits 92 Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number M > 0 there is some number δ > 0 (usually depending on M and a) such that f(x) > M whenever 0 < |x-a| < δ Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number M > 0 there is some number δ > 0 (usually depending on M and a) such that f(x) > M whenever 0 < |x-a| < δ Figure- 36

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Special Limits 93 Example Use the definition of the limit to prove the following limit. Solution Let M > 0 be any number and we’ll need to choose a δ > 0 so that, i.e.

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Special Limits 94 Thus we have

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Special Limits 95 Figure- 37 a a - δ a + δ N f(x) Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number N 0 (usually depending on N and a) such that f(x) < N whenever 0 < |x-a| < δ Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number N 0 (usually depending on N and a) such that f(x) < N whenever 0 < |x-a| < δ

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Special Limits 96 Figure- 38 Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number ε > 0 (however small) there is some number M > 0 (usually depending on ε and a) such that | f(x) - L| M. Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number ε > 0 (however small) there is some number M > 0 (usually depending on ε and a) such that | f(x) - L| M.

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Special Limits 97 Figure- 39

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Special Limits 98 Figure- 40 Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number ε > 0 (however small) there is some number N < 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever x < N. Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number ε > 0 (however small) there is some number N < 0 (usually depending on ε and a) such that | f(x) - L| < ε whenever x < N. f(x) N x y L + ε L L - ε

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Special Limits 99 Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number N > 0 there is some number M > 0 (usually depending on N ) such that f(x) > N whenever x > M. Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number N > 0 there is some number M > 0 (usually depending on N ) such that f(x) > N whenever x > M. N M y x f(x) Figure- 41

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Special Limits 100 If wE say that limit of f(x) does not exist as x tends to a. These limits exist in the extended real line. If wE say that limit of f(x) does not exist as x tends to a. These limits exist in the extended real line.

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Special Limits Properties of limit of functions : 101

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Special Limits Properties of limit of functions : 102

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Special Limits 103

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Special Limits 104 Figure- 42 Present a geometric proof of the following limit. Proof Construct a circle with center at O and radius OA = OD = 1, as in Figure 33. Choose point B on OA extended and point C on OD so that lines BD and AC ar perpendicular to OD. It is geometrically evident that Area of triangle OAC < Area of sector OAD < Area of triangle OBD that is,

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Outline Functions and its graphs. One-one, Onto and inverse functions. Principal values. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 105

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Continuous Function 106 A function f(x) is said to be continuous x = a, if for every neighborhood N ε (f(a)) there is some neighborhood N δ (a) such that if x ϵ N δ (a) then f(x) ϵ N ε (f(a)). A function f(x) is said to be continuous x = a, if (i) f(x) is well defined at x = a, and (ii) A function f(x) is said to be continuous x = a, if (i) f(x) is well defined at x = a, and (ii) A function f(x) is said to be continuous x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) – f(a)| < ε whenever |x-a| < δ A function f(x) is said to be continuous x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) – f(a)| < ε whenever |x-a| < δ

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Continuous Function 107 Figure- 43 f(x) f(a) f(a) - ε f(a) + ε a a - δa + δ

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Continuous Function 108

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Continuous Function 109

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Continuous Function 110 Figure- 44 Figure- 45

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Outline Functions and its graphs. One-one, Onto and inverse functions. Principal values. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 111

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Right hand continuity 112 Or equivalently if for every neighborhood N ε (f(a)) there is some number δ > 0 such that if x ϵ [a, a + δ) then f(x) ϵ N ε (f(a)). Or equivalently if for every neighborhood N ε (f(a)) there is some number δ > 0 such that if x ϵ [a, a + δ) then f(x) ϵ N ε (f(a)). Let f(x) be a function defined on an interval that contains x = a. Then f(x) is said to be right hand continuous or right continuous at x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - f(a) | < ε whenever 0 ≤ x-a < δ (Or a ≤ x < a + δ) Let f(x) be a function defined on an interval that contains x = a. Then f(x) is said to be right hand continuous or right continuous at x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - f(a) | < ε whenever 0 ≤ x-a < δ (Or a ≤ x < a + δ)

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Right hand continuity 113 Figure- 46 f(x) f(a) f(a) - ε f(a) + ε a a + δ

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Right hand continuity 114 Example Figure- 47 H(t) is right continuous at t = 0.

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Left hand continuity 115 Or equivalently if for every neighborhood N ε (f(a)) there is some number δ > 0 such that if x ϵ (a - δ, a ] then f(x) ϵ N ε (f(a)). Or equivalently if for every neighborhood N ε (f(a)) there is some number δ > 0 such that if x ϵ (a - δ, a ] then f(x) ϵ N ε (f(a)). Let f(x) be a function defined on an interval that contains x = a. Then f(x) is said to be left hand continuous or left continuous at x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - f(a) | < ε whenever – δ < x-a ≤ 0 (Or a - δ < x ≤ a) Let f(x) be a function defined on an interval that contains x = a. Then f(x) is said to be left hand continuous or left continuous at x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) - f(a) | < ε whenever – δ < x-a ≤ 0 (Or a - δ < x ≤ a)

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Left hand continuity 116 Figure- 48 f(x) f(a) f(a) - ε f(a) + ε a a - δ

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Left hand continuity 117 Figure- 49 f(a) f(x) is left continuous at x = a.

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Continuous Functions 118 (1) f(x) is defined at all points inside a neighborhood of the point a. (2) Left hand limit and right hand limit of f(x) exist at x = a and both have equal values, i.e. f(a + ) = f(a - ) = L, Where f(a + ) and f(a - ) denote the left hand limit and the right hand limit of f(x) at x = a respectively. (3) The value of f(x) at x = a is equal to L, i.e. f(a) = L. To show continuity of f (x) at x = a, the following conditions must be verified: If any of the above conditions is violated, then f(x) is said to be discontinuous at x = a.

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Continuous Functions 119 There are two kinds of discontinuity. (1) f(x) has discontinuity of the first kind at x = a if f(a + ) and f(a - ) exist, but at least one of them is different from f(a). (2) f(x) has discontinuity of the second kind at x = a if at least one of f(a + ) and f(a - ) does not exist. There are two types of the Discontinuity of the first kind. (i) Solvable discontinuity: if f(a + ) = f(a - ). (ii) Jump discontinuity: if f(a + ) ≠ f(a - ).

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Continuous Functions 120 Figure- 50 f(2) = 4 Solvable discontinuity

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Continuous Functions 121 Example Figure- 51 Jump discontinuity

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Continuous Functions 122 Example Figure- 52 Jump discontinuity

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Continuous Functions 123

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Continuous Functions Example Dirichlet Function The Dirichlet Function d : R → R is defined as follows: d(x) = 1 if x is rational 0 if x is irrational 0 Y X 1 Figure-53 Graph of the function d(x) = 1 if x is rational 0 if x is irrational 124 Discontinuity of the second kind

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Continuous Functions Example Modification-1 of Dirichlet Function The Modification-1 of Dirichlet Function f : R → R is defined as follows: f(x) = x. d(x) x if x is rational 0 if x is irrational Figure-54 Graph of the functionf(x) = x if x is rational 0 if x is irrational 125 This function is continuous at only x = 0.

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Continuous Functions Example Modification-2 of Dirichlet Function The Modification-2 of Dirichlet Function f : R → R is defined as follows: f(x) = x 2. d(x) x 2 if x is rational 0 if x is irrational Figure-55 Graph of the functionf(x) = x 2 if x is rational 0 if x is irrational 126 This function is continuous at only x = 0.

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Continuous Functions Example Modification-3 of Dirichlet Function. It is also known as ruler function. The Modification-3 of Dirichlet Function f : R → R is defined as follows: Figure-56 Graph of the ruler function. 127 This function is continuous at every irrational numbers.

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Continuous Functions 128

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Continuous Functions 129

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Outline Functions and its graphs. One-one, Onto and inverse functions. Principal values. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 130

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Sectional continuity or piecewise continuity 131 Figure- 57

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Outline Functions and its graphs. One-one, Onto and inverse functions. Principal values. Transcendental functions. Bounded and monotonic functions. Limits of functions. Right and left hand limits. Special limits. Continuity. Right and left hand continuity. Sectional continuity. Uniform continuity, Lipschitz continuity. 132

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Uniform Continuity 133 A function f(x) is said to be uniformly continuous in an interval if for every number ε > 0 (however small) there is some number δ > 0 (depending on ε ) such that | f(x 1 ) – f(x 2 )| < ε whenever |x 1 -x 2 | < δ, where x 1 and x 2 are any two points in the interval. A function f(x) is said to be uniformly continuous in an interval if for every number ε > 0 (however small) there is some number δ > 0 (depending on ε ) such that | f(x 1 ) – f(x 2 )| < ε whenever |x 1 -x 2 | < δ, where x 1 and x 2 are any two points in the interval. Let f(x) be continuous in an interval. Then, by definition, at each point a of the interval and for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) – f(a)| < ε whenever |x-a| < δ. If we can find δ for each ε which holds for all points of the interval (i.e. δ depends only one ε and not on a) we say that is uniformly continuous in the interval. Let f(x) be continuous in an interval. Then, by definition, at each point a of the interval and for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that | f(x) – f(a)| < ε whenever |x-a| < δ. If we can find δ for each ε which holds for all points of the interval (i.e. δ depends only one ε and not on a) we say that is uniformly continuous in the interval.

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Uniform Continuity 134 Figure- 58 f(x) f(x 1 ) ε δ x2x2 x1x1 f(x 2 ) a b

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Uniform Continuity 135 Problem Prove that f (x) = x 2 is uniformly continuous in 0 < x < 1. Problem Prove that f (x) = x 2 is uniformly continuous in 0 < x < 1. Theorem If f is continuous in a closed interval, it is uniformly continuous in the interval. Theorem If f is continuous in a closed interval, it is uniformly continuous in the interval. Solution We must show that, given any ε > 0, we can find δ > 0 such that | x 1 2 – x 2 2 | < ε whenever |x 1 -x 2 | < δ, where x 1, x 2 ϵ (0, 1). Now | x 1 2 – x 2 2 )| = |x 1 + x 2 | |x 1 -x 2 | ≤ (| x 1 | + | x 2 |) < (1 + 1) |x 1 -x 2 | =2 |x 1 -x 2 | Therefore given ε > 0 let δ = ε /2. Then if |x 1 -x 2 | < δ, then | x 1 2 – x 2 2 | < ε. Solution We must show that, given any ε > 0, we can find δ > 0 such that | x 1 2 – x 2 2 | < ε whenever |x 1 -x 2 | < δ, where x 1, x 2 ϵ (0, 1). Now | x 1 2 – x 2 2 )| = |x 1 + x 2 | |x 1 -x 2 | ≤ (| x 1 | + | x 2 |) < (1 + 1) |x 1 -x 2 | =2 |x 1 -x 2 | Therefore given ε > 0 let δ = ε /2. Then if |x 1 -x 2 | < δ, then | x 1 2 – x 2 2 | < ε.

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Uniform Continuity 136 Problem Prove that f (x) = x 2 is not uniformly continuous in R = (- ∞, ∞ ). Problem Prove that f (x) = x 2 is not uniformly continuous in R = (- ∞, ∞ ). Solution Suppose that f (x) = x 2 is uniformly continuous in R, then for all ε > 0, there would exist a δ > 0 such that | x 1 2 – x 2 2 | < ε whenever |x 1 -x 2 | < δ where x 1, x 2 ϵ (0, 1). Take x 1 > 0 and let x 2 = x 1 + δ/2. Write ε ≥ | x 1 2 – x 2 2 )| = |x 1 + x 2 | |x 1 -x 2 | = (2 x 1 + δ/2 ) δ/2 > x 1 δ Therefore x 1 ≤ ε/δ for all x 1 > 0, which is a contradiction. It follows that f (x) = x 2 cannot be uniformly continuous in R = (- ∞, ∞ ). Solution Suppose that f (x) = x 2 is uniformly continuous in R, then for all ε > 0, there would exist a δ > 0 such that | x 1 2 – x 2 2 | < ε whenever |x 1 -x 2 | < δ where x 1, x 2 ϵ (0, 1). Take x 1 > 0 and let x 2 = x 1 + δ/2. Write ε ≥ | x 1 2 – x 2 2 )| = |x 1 + x 2 | |x 1 -x 2 | = (2 x 1 + δ/2 ) δ/2 > x 1 δ Therefore x 1 ≤ ε/δ for all x 1 > 0, which is a contradiction. It follows that f (x) = x 2 cannot be uniformly continuous in R = (- ∞, ∞ ).

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Uniform Continuity 137 Problem Prove that f (x) = 1/x is not uniformly continuous in 0 < x < 1. Problem Prove that f (x) = 1/x is not uniformly continuous in 0 < x < 1. Solution Suppose that f (x) = 1/x is uniformly continuous in R, then for all ε > 0, there would exist a δ > 0 such that | 1/x 1 – 1/x 2 | < ε whenever |x 1 -x 2 | < δ where x 1, x 2 ϵ (0, 1). ε > | 1/x 1 – 1/x 2 | = (|x 2 -x 1 | )/ (x 1 x 2 ) Or |x 1 -x 2 | < x 1 x 2 ε Therefore, to satisfy the definition of uniform continuity we would have to have δ ≤ x 1 x 2 ε for all x 1, x 2 ϵ (0, 1), but that would mean that δ ≤ 0. Therefore there is no single δ > 0. Therefore f (x) = 1/x is not uniformly continuous in 0 < x < 1. Solution Suppose that f (x) = 1/x is uniformly continuous in R, then for all ε > 0, there would exist a δ > 0 such that | 1/x 1 – 1/x 2 | < ε whenever |x 1 -x 2 | < δ where x 1, x 2 ϵ (0, 1). ε > | 1/x 1 – 1/x 2 | = (|x 2 -x 1 | )/ (x 1 x 2 ) Or |x 1 -x 2 | < x 1 x 2 ε Therefore, to satisfy the definition of uniform continuity we would have to have δ ≤ x 1 x 2 ε for all x 1, x 2 ϵ (0, 1), but that would mean that δ ≤ 0. Therefore there is no single δ > 0. Therefore f (x) = 1/x is not uniformly continuous in 0 < x < 1.

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Lipschitz continuity 138

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Lipschitz continuity 139

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Lipschitz continuity 140

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Lipschitz continuity 141 since

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Lipschitz continuity 142 since

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143 Thank you all

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LIMITS AND DERIVATIVES 2. We noticed in Section 2.3 that the limit of a function as x approaches a can often be found simply by calculating the value.

LIMITS AND DERIVATIVES 2. We noticed in Section 2.3 that the limit of a function as x approaches a can often be found simply by calculating the value.

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