real analysis - Questions relating differentiability to continuity and integrability

These questions are from Stephen Abbott's "Understanding Analysis", 7.5.2, following a brief explanation of the Fundamental Theorem of Calculus. I have ideas but I feel like I need help still.

Decide whether each statement is true or false, providing a short justification for each conclusion.

(a). If $h'=g$ on $[a,b]$, then $g$ is continuous on $[a,b]$

If I understand the question correctly, it's asking that if a function is differentiable on $[a,b]$ then the derivative has to be continuous on that interval. I have seen a counter example to this from previous questions, which is $f(x)=\{x^2 sin(1/x), x \neq 0$,and $0, x=0 \}$

(b) If $g$ is continuous on $[a,b]$, then $g=h'$ for some $h$ on $[a,b]$

Is this asking if a function is continuous does that mean it's differentiable? Then the answer is no because $f(x)=|x|$ is not differentiable at $x=0$. But I'm not sure if I'm even reading the problem correctly.

(c) If $H(x) = \int_{a}^{x}h$ is differentiable at $c \in [a,b], $ then $h$ is continuous at $c$.

Intuition tells me yes. I can integrate a step function, for example. The result is continuous but not differentiable.

Any input or hints would be appreciated.

Answer

Hint for (c). Consider the function $h$ which is $1$ at $0$ and it is $0$ otherwise.

What is $H(x) = \int_{-1}^{x}h(t) dt$ for $x\in [-1,1]$?
Is $H$ differentiable at $0$?

P.S. (a) is false by your counterexample. (b) is true, see MathematicsStudent1122's comment.