1. CONTINUITY

2. A function f(x) is continuous at a number a if: 3 REQUIREMENTS f(a) exists – a is in the domain of f exists

3. If a function is continuous, you can draw or trace the graph without lifting your pencil. If a function is not continuous, it is said to be discontinuous.

4. Example Iscontinuous? No, 3 is not in the domain of f –The function is discontinuous at x = 3 Go to Graph Question

5. Back to Problem

6. How could we make the function continuous? Rename a point at x = 3 This removes the discontinuity Hence, the discontinuity is REMOVABLE

7. Example Discontinuous at x = 0 Is there any way to remove the discontinuity? NO INFINITE Discontinuity

8. Discontinuous at n Є Z JUMP Discontinuity Discontinuous from the left but not from the right

9. Example Determine if the function is continuous.

10. Intermediate Value Theorem If f is continuous on [a,b] and N is any number strictly between f(a) and f(b), then there exists a number c in (a,b) such that f(c)=N Ex: Show that there is a root of the equation between 1 and 2.