Continuity & IVT

1. Continuity & IVT Section 1.4

2. Continuity at a Point • A function f is continuous at c if the following three conditions are met. • f(c) is defined • exists

3. Continuity on an open interval • A function is continuous on an open interval (a, b) if it is continuous at each point in the interval.

4. Continuity on a closed interval • A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and and

5. Discontinuities • Removable Discontinuity • A discontinuity is considered removable if a hole exists in the graph. To create this hole exists, however it does not equal f(c) or f(c) does not exist.

6. Discontinuities • Nonremovable Discontinuities • A discontinuity is considered nonremovable if does not exist

7. Graph and identify where the discontinuities occur

8. Intermediate Value Theorem • If f is continuous on the closed interval [a, b] and y is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = y. • The IVT does not tell you the value of c, only that it exists.

9. Use the IVT to show that f(x) = x2 + 2x – 1 has a zero in the interval [0, 1]