I'll just respond to the second part...

You're asked to prove that if F is a function from R->R, that

.

If f is differentiable at a, by the limit definition of a derivative this means that

I don't know how detailed your proof is expected to be, but I would say that it follows from this limit definition of the derivative that

. To be continuous at a certain point, the limit must exist at a, and f(a) must be defined (meaning no holes or something like that). Thus again by the limit above, f(a) must exist otherwise the derivative would not.