I want to show the following statement.
Let $f$ be of bounded variation on $\mathbb{R}$. Let $$V_f(x)=V(f; -\infty, x)$$ Prove that $f$ is continuous at $c$ if and only if $V_f$ is continuous at $c$. (Douglass, 5.25)
I can prove this fact for $V_f(x)=V(f; a, x)$, when the bounded variation is over the interval $[a, b]$. However, I am having a hard time for proving when one of the interval bound is negative infinity. Douglass defines the definition of these kinds of bounded variation as $$V(f; -\infty, x)=\sup\{V(f; a, b) | [a, b] \subset (\infty, x]\}$$ So I have two supremum in my definition of $V_f(x)$. How can I prove this statement?