Fundamental Theorem 1 & 2

Hi, the way I understand it, the first part of the fundamental theorem of calculus says that every function that is continuous on the closed interval has an antiderivative. If we denote the antiderivative by it looks like,

.

Now, F is continuous on and differentiable on , and for all in .

*Why is not differentiable on the closed interval ?*

A corollary of this is that

From what I read, the second part is almost the same as the first part but it does not assume continuity, but it assumes that an antiderivative of exists.

*Does this mean that every continuous function has an antiderivative (part 1), but if a function has an antiderivative then it must not necessarily be continuous?*

I do not clearly see the difference between the corollary of part one, and theorem part 2.

Thanks!