F is not (necessarily) differentiable at the end points, a and b, because that would require that the limits of the difference quotient from both directions be the same. And we have no information about F for x> b or for x< a.

Yes, if a function is continuous, then it has an anti-derivative. But that is NOT an "if and only if" theorem. There exist non-continuous functions that have anti-derivatives. In fact, take any "jump function", integrate it, and you get an example.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?

If, say, f(x)= 0 for x< 0, f(x)= 1 for , then the antiderivative function is F(x)= 0 for x< 0, F(x)= x for .

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

Thanks!