# Relationship between continuity and differentiability of Riemann integrals [Basic real analysis]

I'm a bit puzzled by some of the conditions on integrability, differentiability, and continuity. I am working on the below question(s), and would like to see if my intuition checks out. As this is a homework question, I'd prefer hints over straight answers.

Let $f$ be [Riemann] integrable on $[a,b]$, and $c,x \in (a,b)$. Define $$F(x) = \int_a^xf$$ Then give proofs or counterexamples to the following:

a) If $f$ is differentiable at $c$, then so is $F$

b) If $f$ is differentiable at $c$, then $F^{\prime}$ is continuous at $c$

c) If $f^{\prime}$ is continuous at $c$, then $F^{\prime}$ is differentiable at $c$

Ok here are my thoughts:

a) I think I can boil this one down to the Fundamental Theorem of Calculus. Since $F(x)$ is an antiderivative, it must be differentiable.