Continuity, the Intermediate Value Theorem, and the Bisection Method

1. Continuity, the Intermediate Value Theorem, and the Bisection Method

2. Continuity, IVT & Bisection Method A function is continuous at a point cif and only if a. is defined b. exists c. the two are equal; i.e.

3. Continuity, IVT & Bisection Method A function is continuous on an open interval (a, b) if and only if is continuous at all points on the open interval (a, b) A function is continuous on a closed interval [a, b] if and only if is continuous on the open interval (a, b) and the left and right hand limits and exist and equal at the endpoints

4. Continuity, IVT & Bisection Method Intermediate Value Theorem: If is continuous on the closed interval [a, b] and if L is a value between and then there is a point c between a and b such that

5. Continuity, IVT & Bisection Method Bisection Method: Let be continuous on the closed interval [a, b] and let the two endpoints and have opposite signs. Compute where c’ is the midpoint If and have opposite signs then a zero is between a and c’; otherwise a zero is between c’ and b. Repeat