Suppose $f\in C^1$. Define
$$g(x) = \begin{cases} f'(x) & \text{if $x\ne x_0$} \\ 0 & \text{if $x=x_0$} \end{cases}$$
(a) If $f$ is continuous at $x=x_0$, show that $f'=g$ in the sense of distributions
(b) If $f$ has a jump discontinuity at $x_0$ where
$$a=\lim\limits_{x \to x_0^-} f(x)$$ $$b=\lim\limits_{x \to x_0^+} f(x)$$
show that $f'=g+(b-a)\delta_{x_0}$ in the sense of distributions
I'm trying to proceed using definitions of distributional derivatives, continuity, and the delta function, but I can barely make sense of what the question is suggesting.
Any help is greatly appreciated!