1. Section 1.4: Continuity and One-Sided Limits

2. Example
Use the graph of t(x) to determine the intervals on which the function is continuous.

3. Example 2
Sketch a graph of the function with the following characteristics:
Continuous on the interval: [-7,4)U(4,6)U(6,7) and Range: (-∞,10)
exists, Continuous on the interval: (-∞,-3)U(-3,∞).

4. When x=5, all three pieces must have a limit of 8.
Example 3
Find values of a and b that makes f(x) continuous.
When x=5, all three pieces must have a limit of 8.

5. For every question of this type, you need (1), (2), (3), conclusion.
Continuity at a Point
A function f is continuous at c if the following three conditions are met:
is defined.
exists. c L
f(x) x
For every question of this type, you need (1), (2), (3), conclusion.

6. Example 1 Show is continuous at x = 0. f is continuous at x = 0
The function is clearly defined at x= 0
With direct substitution the limit clearly exists at x=0
The value of the function clearly equals the limit at x=0
f is continuous at x = 0

7. Example 2 Show is not continuous at x = 2.
The function is clearly 10 at x = 2
With direct substitution the limit clearly exists at x=0
The behavior as x approaches 2 is dictated by 8x-1
The value of the function clearly does not equal the limit at x=2
f is not continuous at x = 2

8. Discontinuity
If f is not continuous at a, we say f is discontinuous at a, or f has a discontinuity at a.
Types Of Discontinuities
Removable
Able to remove the “hole” by defining f at one point
Non-Removable
NOT able to remove the “hole” by defining f at one point
Typically a hole in the curve
Step/Gap
Asymptote

9. Example
Find the x-value(s) at which is not continuous. Which of the discontinuities are removable?
If f can be reduced, then the discontinuity is removable:
This is the same function as f except at x=-3
There is a discontinuity at x=-3 because this makes the denominator zero.
f has a removable discontinuity at x = 0

10. One-Sided Limits: Left-Hand
If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from values less than c, the left-hand limit is L. c L
f(x) x
The limit of f(x)…
is L.
Notation:
as x approaches c from the left…

11. One-Sided Limits: Right-Hand
If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from values greater than c, the right-hand limit is L. c L
f(x) x
The limit of f(x)…
is L.
Notation:
as x approaches c from the right…

12. Example 1
Evaluate the following limits for

13. Example 2
Sketch a graph of a function with the following characteristics:

14. Example 3
Analytically find

15. The Existence of a Limit
Let f be a function and let c be real numbers. The limit of f(x) as x approaches c is L if and only if c L
f(x) x

16. You must use the piecewise equation:
Example 1
Analytically show that
Use when x>2
Use when x<2
You must use the piecewise equation:

17. You must use the piecewise equation:
Example2
Analytically show that is continuous at x = -1.
You must use the piecewise equation:
Use when x>-1
Use when x<-1

18. Continuity on a Closed Interval
A function f is continuous on [a, b] if it is continuous on (a, b) and a
f(a)
f(b) x b

19. Is the middle is continuous?
Example
Discuss the the continuity of
Are the one-sided limits of the endpoints equal to the functional value?
By direct substitution:
The domain of f is [-1,1]. From our limit properties, we can say it is continuous on (-1,1)
f is continuous on [-1,1]
Is the middle is continuous?

20. Properties of Continuity
If b is a real number and f and g are continuous at x = c, then following functions are also continuous at c:
Scalar Multiple:
Sum/Difference:
Product:
Quotient: if
Composition:
Example: Since are continuous, is continuous too.

21. Intermediate Value Theorem
If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that: a b
f(a)
f(b) k c

22. Example
Use the intermediate value theorem to show has at least one root.
Find an output less than zero
Find an output greater than zero
Since f(0) < 0 and f(2) > 0
There must be some c such that f(0) = 0 by the IVT
The IVT can be used since f is continuous on [-∞,∞].

23. Example Show that has at least one solution on the interval .
Solve the equation for zero.
Find an output greater than zero
Find an output less than zero
Since and
The IVT can be used since the left and right side are both continuous on [-∞,∞].
There must be some c such that cos(c) = c3 - c by the IVT