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 $\displaystyle [a,b]$ has an antiderivative. If we denote the antiderivative by $\displaystyle F$ it looks like,

$\displaystyle F(x) = \int^x_a f(t)dt$.

Now, F is continuous on $\displaystyle [a,b]$ and differentiable on $\displaystyle (a,b)$, and $\displaystyle F'(x)=f(x)$ for all $\displaystyle x$ in $\displaystyle (a,b)$.

*Why is $\displaystyle F$ not differentiable on the closed interval $\displaystyle [a,b]$?*

A corollary of this is that

$\displaystyle \int^b_a f(x)dx = F(b)-F(a).$

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 $\displaystyle f$ 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!