1.4 Continuity and One-Sided Limits

1. 1.4 Continuity and One-Sided Limits This will test the “Limits” of your brain!

2. Definition of Continuity A function is called continuous at c if the following three conditions are met: 1. f(c) is defined 2. 3. A function is continuous on an open interval (a,b) if it is continuous at each point in the interval.

3. Two Types of Discontinuities • Removable • Point Discontinuity • Non-removable • Jump and Infinite the open circle can be filled in to make it continuous Removable example

4. Non-removable discontinuity. Ex. -1 1

5. Determine whether the following functions are continuous on the given interval. yes, it is continuous ( ) 1

6. ( ) discontinuous at x = 1 removable discontinuity since filling in (1,2) would make it continuous.

7. yes, it is continuous

8. One-sided Limits Limit from the right Limit from the left Find the following limits 0 D.N.E. 1 D.N.E.

9. Step Functions “Jump” Greatest Integer -1 0 D.N.E.

10. g(x)= Is g(x) continuous at x = 2? 3 3 g(x) is continuous at x = 2

11. 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 f(a) In this case, how many c’s are there where f(c) = k? k f(b) 3 [ ] a b

12. Show that f(x) = x3 + 2x –1 has a zero on [0,1]. f(0) = 03 + 2(0) – 1 = -1 f(1) = 13 + 2(1) – 1 = 2 Since f(0) < 0 and f(1) > 0, there must be a zero (x-intercept) between [0,1].