Statement. Suppose is a function defined on a closed interval (with ) satisfying the following three conditions:. is a continuous function on the closed interval .In particular, is (two-sided) continuous at every point in the open interval, right continuous at, and left continuous at. is differentiable on the open interval, i.e., the derivative of exists at all points in the open interval.
Rolle's theorem is one of the foundational theorems in differential calculus. It is a special case of, and in fact is equivalent to, the mean value theorem, which in turn is an essential ingredient in the proof of the fundamental theorem of calculus. The theorem states as follows: A graphical demonstration of this will help our understanding; actually, you'll feel that it's very apparent: In.