Continuous and differentiable function in Rolle's theorem

Question

When we state Rolle's theorem, we say that the function is continuous on a closed interval and differentiable in an open interval. Why we do not assume differentiability on the closed interval?

Answer 1

Because differentiation is not well defined on a closed interval. In fact we need to know what happen on the right and on the left of all point of the interval.

Hence in order to prove that a continuous function admits maximum or minimum we do not need the differentiability at the extreme points of the interval.

Answer 2

Also: because even if we define differentiability in the border points as existence of lateral derivatives, in the proof of the theorem only derivatives in the interior are required. Sketch of proof:

If $f$ is constant, the conclusion is trivial.

If $f$ isn't constant, max or min are reached in the interior of the interval and this global max/min is also local max/min. Then, derivative in this point is zero.