3.2: Continuity

1. 3.2: Continuity Objectives: To determine whether a function is continuous Determine points of discontinuity Determine types of discontinuity Apply the Intermediate Value Theorem

2. CONTINUITY AT A POINT—No holes, jumps or gaps!! A function is continuous at a point c if: 1.) f(c) is defined 2.) exists 3.) = f(c) A FUNCTION NEED NOT BE CONTINUOUS OVER ALL REALS TO BE A CONTINUOUS FUNCTION

4. Does f(2) exist? Does exist? Does exist? Is f(x)continuous at x = 2? Do the same for x= 1, 3, and 4.

5. Removable discontinuity(hole in graph) • Limit exists at c but f(c)≠ the limit • Can be fixed. Set f(c) = • This is called a continuous extension

6. Example: Find the values of x where the function is discontinuous. State if it is a removable discontinuity. If so, fix it.

7. OTHER TYPES OF DISCONTINUITY • JUMP: ( RHL ≠ LHL) • INFINITE: • OSCILLATING

8. Continuity on a closed interval A function is continuous on a closed interval [a,b] if: • It is continuous on the open interval (a,b) • It is continuous from the right at x=a: • It is continuous from the left at x=b:

9. example IT IS CONTINUOUS ON ITS DOMAIN. But discontinuous on x values not in the domain.

10. Key Functions and where they are continuous

11. A continuous function is one that is continuous at every point in its domain. It need not be continuous on all reals. Where are the functions discontinuous? If it is removable discontinuity, fix it!!

12. For piecewise functions… • Check to make sure each “piece” is continuous • Check the x values where it changes functions. Remember, the following must be true to be continuous at x:

13. Where are the functions discontinuous? Fix removable!