Suppose that $f:[a, b] \to \mathbb{R}$ is a function of bounded variation. Define $g:[a, b] \to \mathbb{R}$ by $g(x) = V_a^x f$. Show that $f$ is continuous at $x \in [a, b]$ iff $g$ is continuous at $x$.
The converse is simple. I'm not sure how to prove the forward implication.
I was wondering if I could get a hint?
Thanks!