1. Continuity
A function is Continuous if it can be drawn without lifting the pencil, or writing utensil, from the paper.
A continuous function has no
breaks,
holes,
or jumps
in its graph.
Three conditions exist for which the graph of f is not continuous at x = c.
( | )
a c b
( | )
a c b
( | )
a c b
The limit of f(x) exists at x = c, but is not equal to f(c).
The function is not defined at x = c.
The limit of f(x) does not exist at x = c.

2. Definition of Continuity
This is very important, so pay attention.
Continuity at a Point:
A function is continuous at c if the following three conditions are met.
Continuity on a Open Interval:
A function is continuous on an open interval (a, b) if it is continuous at each point in the interval. A function that is continuous on the entire real line
This stuff gives me a headache.
is everywhere continuous.

3. Removable and Non-Removable Discontinuity
Discontinuities fall into two categories: Removable and Non-Removable.
A discontinuity at c is removable if f can be made continuous by appropriately defining or redefining f(c).
If defining or redefining f(c) still does not make f continuous, then the discontinuity is non-removable.
( | )
a c b
( | )
a c b
( | )
a c b
Removable
Discontinuity
Removable
Discontinuity
Non-Removable
Discontinuity

4. Continuity Examples That was easy
f(x) is not defined at x = 2, therefore, by definition, f(x) is not continuous at x = 2.
does not exist, therefore, by definition, f(x) is not continuous at x = 2.
Since f(x) is undefined at x = 2, we can redefine f(2) and make f(x) continuous.
The discontinuity is non-removable.
Since by redefiningf(2) we would still not make f(x) continuous.
Therefore the discontinuity is removable.
f(x) is defined along the entire real line, therefore, by definition, f(x) is everywhere continuous.
Therefore, by definition, f(x) is not continuous at x = 2.
If the point (2, 2) is removed & replaced with the point (2, 1), f(x) would be continuous.
That was easy
Therefore the discontinuity is removable.

5. Discussing Continuity
Discuss the continuity of each of the following functions.
Since f(0) results in an undefined function, this function is discontinuous at x = 0.
It seems like this function is continuous on the entire real line.
Since there is no way to define f(0) so as to make the function continuous at x = 0, the function has non-removable discontinuity at x = 0.
Since there are no holes, breaks, or jumps in the graph, that means the function is everywhere continuous.

6. More Discussing Continuity
Discuss the continuity of each of the following functions.
Since f(1) results in an undefined function, this function is discontinuous at x = 1.
It seems like this function is continuous on the entire real line.
Since the function can be redefined at f(1), the function has removable discontinuity at x = 0.
Since there are no holes, breaks, or jumps in the graph, that means the function is everywhere continuous.

7. Discuss Continuity Only
Homework
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Do Not Find Limits
Discuss Continuity Only
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8. One Sided Limits
The limit of a function can be evaluated from the right or from the left.
A limit from the left means that x approaches c from the left side, which means values that are less than c.
A limit from the right means that x approaches c from the right side, which means values that are greater than c.
A limit from the left side is denoted as:
A limit from the right side is denoted as:

9. Take a look at what that really means.
Existence of a Limit
Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if
and
Take a look at what that really means.
Since the limit is not equal to the same number from the left and from the right, the limit does not exist.

10. Finding One Sided Limits
Find the following limits if they exist.

11. Using Limits to Prove Discontinuity
Use the definition of continuity to prove the following function is discontinuous at x = 5.
We must look at this as two separate problems
and
As x approaches 5 from the left, the value of x will be less than 5, therefore we can use the following inequality.
As x approaches 5 from the right, the value of x will be greater than 5, therefore we can use the following inequality.
Since the absolute value of a positive number is positive, and a positive divided by a positive is positive, this leads us to:
Since the absolute value of a negative number is positive, and a positive divided by a negative is negative, this leads us to:
Since the limit is not the same from the left as it is from the right, the limit does not exist. Since the limit does not exist at x = 5, the function is discontinuous at x = 5.

12. Homework
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Find Limits Only
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13. Continuity on a Closed Interval
A function is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and
and
The function is continuous from the right at a and continuous from the left at b.
We better look at a diagram.
This is also true for functions that are on intervals in the form (a, b] or [a, b).
f(a)
is continuous on the infinite interval
f(b)
[ ]
a b
is continuous on the infinite interval

14. Example of Continuity on a Closed Interval
We better look at a diagram.
Discuss the continuity of
The domain of the function is the closed interval [-1, 1]
Since the function is continuous on the closed interval [a, b] where a = -1 and b = 1, we must check to see if
and

15. Properties of Continuity
If b is a real number, and f and g are continuous at x = c, then the following functions are also continuous at c.
1) Scalar Multiple
3) Product
2) Sum or Difference
4) Quotient
The following types of functions are continuous at every point in their domains.
Polynomial Functions
Radical Functions
Rational Functions
Trigonometric Functions

16. Homework
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17. The Intermediate Value Theorem
If f(x) is a continuous function on the closed interval [a, b], then for every d between f(a) and f(b), there exists a c so that f(c) = d.
That sounds really confusing.
Let me show you how easy it is.
Now I get it. If d is between f(a) and f(b) , then a corresponding c, between a and b, exists so that f(c) = d. I think I should probably push the easy button.
f(b)
f(c) d
f(a)
[ ]
a b c
That was easy

18. Application of The Intermediate Value Theorem
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.
[–1, 5]
Since f is a polynomial function, f is continuous on [–1, 5]
and
(x = –6 is not in the interval)
The Intermediate Value Theorem applies.
Therefore,

19. Another Application of The Intermediate Value Theorem
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.
[2, 7]
Since f is a polynomial function, f is continuous on [2, 7]
and
The Intermediate Value Theorem applies.
(x = –2 is not in the interval)
Therefore,

20. Rational Root Theorem Example
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.
[-1, 4]
Intermediate Value
Theorem applies.
and
Use Synthetic Division to find the other factor.
Use the Rational Root Test
The 2 factors are:
The factors of the last term are:
Has no real roots
Therefore,
The value of c guaranteed by the theorem is 3.

21. Homework
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