1.4 Continuity and One-Sided Limits (Part 2)

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 76 • 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 1.11) 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 non-removable discont at x=1

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 4 1

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. Homework 1.4 (page 77) #25-53 odd, 57, 59, 75, 77, 79, 81, 83, 87 (#79 and 81 use the bisection method) 79. 11/16 81. 9/16