Why the mean value theorem is not differentiable in the close interval?
We'll recall the fact that a continuous function over a closed interval [a,b] is differentiable over the opened interval (a,b).
For a function to be differentiable at the endpoints a and b, it has to be differentiable from both sides of a and it has to be differentiable from both sides of b. But this is impossible, since the domain of definition of the function comprises values larger than a, including a, and values smaller than b, including b.
Therefore, if the domain of definition of the continuous function is [a,b], the function is differentiable from the right at a and from the left at b.
That is why a continuous function, whose domain of definition is the closed interval [a,b], can be differentiable in the opened interval (a,b), only.