1. Calculus 12 Continuity

2. The mathematical definition of continuity is similar to the everyday use of the word –uninterrupted connection or union –the property of a continuous and connected period of time We say that a function is continuous if:

3. This implies three things: (1)The function exists at f(a) (2)The limit exists at f(a) (3)The limit at f(a) equals the function f(a)

4. If a function is not continuous, we say that f(x) is discontinuous at “a” or f(x) has discontinuity There are three statements that must be fulfilled in order for a function to be continuous which implies there are three ways for a function to exhibit discontinuity.

5. Eg (1) f(x) is discontinuous at a=2 because f(2) does not exist The graph would be similar to

6. Eg (1) (2) f(x) is discontinuous at a=0 because the limit does not exist

7. Eg (1) (2) (3)

8. The function is discontinuous at a=2 because the function does not equal the limit. The graph would be similar to

9. Eg (1)f(1) = 2(1)+3 = 5 (2)

10. The function is discontinuous at a=1 because the limit does not exist. The graph would be similar to

11. Eg (1) (2) (3) f(x) is continuous at a=2

12. Homework Stewart Calculus page 94 #1(a) 3-7 12-20 (don’t need to sketch)