1. 1.4 Continuity
Calculus

2. Understanding Continuity
Section 1
Understanding Continuity

3. Most of the techniques of calculus require that functions be continuous.
A function is continuous if you can draw it in one motion without picking up your pencil.

4. Definition of Continuity
A function is continuous at a point c if the following three conditions are met.
f(c) is defined 2. 3.

5. Definition of Continuity
A function is continuous on an open interval (a, b) if it is continuous on each point in the interval. A function that is continuous on the entire real line is everywhere continuous.
f(x) is continuous on (-3,2)

6. One Sided Limits and Continuity
A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a, b) and
The function is continuous from the right at a and continuous from the left at b.
f(x) is continuous on [-3,2]

7. Example 1: Continuity of a Function
Discuss the continuity of each function.

8. Example 1a Solution
There is a removable discontinuity at x = 1 because f(1) is not defined (Definition of Continuity Condition 1).

9. Example 1b Solution The function has a nonremovable
discontinuity at x = 0 because does not exist (Definition of Continuity Condition 2).

10. Example 1c Solution
The limit from the right of x = 2 does not equal the limit from the left. Therefore, the limit as x approaches 2 does not exist.
Function has a discontinuity at x = 2 because does not exist (Definition of Continuity Condition 2).

11. Example 1d Solution
The function is continuous on the entire real line.

12. Try These
Discuss the continuity of each function.

13. Removable Discontinuities:
(You can fill the hole.)
Nonremovable Discontinuities:
infinite
oscillating
jump

14. “Discussing Continuity”
Continuous or discontinuous?
If discontinuous
Removable or nonremovable discontinuity?
At what x-value is the discontinuity?
What condition is not met? (See slide 4 for conditions)
Condition 1: f(c) is defined - means the function exists when x = c
Condition 2: limit as x approaches c of f(x) existis – means that the “two sides of the functions meet at c,” no “jumps, or asymptotes
Condition 3: Limit and value are equal – means the is no “hole” with a “dot” filled in elsewhere

15. Continuity by Function Type
Polynomials are everywhere continuous
Sine and Cosine are everywhere continuous
Rational functions and other trig functions are continuous except at x-values where their denominators equal zero.
“Removable” discontinuity if factoring and canceling “removes” the zero in the denominator
“Non-removable” otherwise. (Recall that vertical asymptotes occur where numerator is nonzero and the denominator is zero.)
Root functions are continuous, except at x-values that would result in a negative value under an even root
For piecewise functions, find the f(x) values at the x-value separating the regions of the function.
If the f(x) values are equal, the function is continuous.
Otherwise, there is a (non-removable) discontinuity at this point.

16. Properties of Continuity
Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.
Also: Composites of continuous functions are continuous.
Examples:
y = 3x + cos(x)

17. Finding the Intervals on Which a Function is Continuous
Section 2
Finding the Intervals on Which a Function is Continuous

18. Examle 2: Describe the interval(s) on which f is continuous.
Solution: The function is continuous on the interval [-3, )

19. Try This: Find the interval on which f is continuous.
Solution: f(x) has a non – removable discontinuity at x = 2.
The intervals on which f is continuous are (- , 2] and [2, )

20. The Intermediate Value Theorem
Section 3
The Intermediate Value Theorem

21. Suppose this pool holds 22,000 gal. of water
Suppose this pool holds 22,000 gal. of water. Then there is some point in time, when it was being filled that it held exactly 15,000 gal. It could not have skipped over that number.

22. Intermediate Value Theorem
If a function is continuous between a and b, then it takes on every value between and
Because the function is continuous, it must take on every y value between f(a) and f(b)

23. Here is a formal statement of the Theorem:

24. Example 3: Intermediate Value Theorem
Without graphing, show that f(x) = 2x4 -3x2 + 5x +2 has at least one zero between -2 and -1.
Solution: This function is continuous on the interval [-2, -1] (and everywhere else for that matter), and
f(-2) =12
f(-1) = -4
Therefore somewhere between
x = -2, and x = -1, f(x) passes through 0.

25. Try This
For the function f(x) = x2 -6x + 8 ,
a) use the Intermediate Value Theorem to show that the function has a zero on the interval [0, 3]
b) find the value of "c" guaranteed by the theorem such that f(c) = 0.

26. Solution f(0) = 8 f(3) = -1 0 is between -1 and 8. f(c) = 0
Verify that the Intermediate Value theorem applies on [0, 3] and find the value of "c" guaranteed by the theorem. f(c) = 0
f(x) = x2 -6x + 8
f(0) = 8
f(3) = -1
0 is between -1 and 8.
f(c) = 0
c2 – 6c + 8 = 0
c = 2 or 4
Not 4, why?