1. Section 1.4 Continuity and One-sided Limits

2. Continuity – a function, f(x), is continuous at x = c only if the following 3 conditions are met: 1. is defined 2. exists 3. Continuous on an open interval (a, b) A function is continuous on (a, b) only if it is continuous at every point in (a, b).

3. Condition 1 is not met: hole in graph c f(c) is not defined

4. Condition 2 is not met: jump or asymptote c does not exist c

5. Condition 3 is not met: hole in graph and function defined elsewhere. c L f(c)

6. Two Types of discontinuities 1.removable – function that can be made continuous at a point by redefining f(c). ex: hole in graph 2.nonremovable – cannot redefine f(c) to make the function continuous. ex: asymptote or jump in graph

7. Examples: Discuss the continuity of each. 1. nonrem. discont. @ x = 0 (asymptote)

8. Examples: Discuss the continuity of each. 2. rem. disc. @ x = 1 (redefine f(1)=0) 1

9. Examples: Discuss the continuity of each. 3. nonrem. disc. @ x = 2 (jump) 2 6 8

10. Examples: Discuss the continuity of each. 4. cont. on (-∞, ∞)

11. One-sided Limits limit from the right limit from the left

12. Examples: Evaluate each limit. 5. = 0 -2

13. Examples: Evaluate each limit. 6. = DNE -2

14. Examples: Evaluate each limit. 7. = 0 1

15. Examples: Evaluate each limit. 8. = -2 -2

16. Existence of a Limit When, then Continuity on a closed interval [a, b] f(X) is continuous on (a, b) and