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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!

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This might help: math.stackexchange.com/questions/144162/… –  Thomas Feb 22 '13 at 15:52

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