1. 1.4 Continuity and One-Sided Limits (Part 2)

2. Objectives
Determine continuity at a point and continuity on an open interval.
Use properties of continuity
Understand and use the Intermediate Value Theorem.

3. Discontinuity
A function is discontinuous if:

4. Continuity
A function is continuous at c if:
f(c) is defined

5. Continuous on (a,b)
A function is continuous on (a,b) if it is continuous at each point in the interval.
If the function is continuous on (-∞,∞), it is everywhere continuous.

6. Categories of Discontinuities
Removable (you can redefine f(c) to make f continuous)
Non-removable (limit doesn't exist at c)

7. Example 1
non-removable
removable
continuous
continuous

8. Page 78
Look at problems 1-6 and discuss removable and non-removable discontinuities.

9. Continuous on a Closed Interval
f is continuous on [a,b] if it is continuous on (a,b) and

10. Example
Therefore, f(x) is continuous on [-1,1].

11. Properties of Continuity (Theorem 1.1)
If f and g are continuous at x=c, then the following functions are also continuous at c:
bf (where b is a real number)
f±g (sum and difference)
fg (product)
f/g (quotient) (g≠0)

12. Continuous Functions
These functions are continuous at every point in the domain:
polynomial functions
rational functions
radical functions
trig functions

13. Continuity of Composite Functions
If g is continuous at c and f is continuous at g(c), then (f◦g)(x)=f(g(x)) is continuous at c.
Since 3x is cont everywhere and since sinx is cont everywhere, sin(3x) is continuous everywhere.
Since x2+1 is cont everywhere and since √x is cont everywhere in its domain, √ x2+1 is continuous.

14. Example removable discont at x=1 non-removable discont at x=2

15. Intermediate Value Theorem
If f is continuous on [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 f(c)=K.

16. Example
If you know that f(1)=2 and f(4)=5, and f is continuous, then there has to be a number c in (1,4) where f(c)=3. 5 3 2 1 4

17. Locating Zeros
You can use the I.V.T. to help narrow down the location of zeros.
If you know a function is continuous, and f(a)<0 and f(b)>0, then there has to be at least one zero in (a,b).

18. Bisection Method
The Bisection Method is used to approximate zeros (roots).
Start with an interval where f(a) and f(b) have different signs.
Evaluate the midpoint of [a,b], and use it to bisect the interval.
Keep evaluating the midpoint of each new interval and bisecting until the required accuracy is reached.

19. Example
Let f(x)=x5+x3+x2-1. Use the bisection method to find a number in [0,1] that approximates a zero of f with an error <1/16.

20. (#87 and 89 use the bisection method)
Homework
1.4 (page 79)
#27-53 odd,
65, 67, 83,
87, 89, 91
(#87 and 89 use the bisection method)