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 $c$.
Update: $V(f ;-\infty, c)=\sup \{V(f: a, b):[a, b] \subset(-\infty, c]\}$
It seems that the proof for continuity of function $f$ at closed interval $[a,b]$ can't be applied for the interval $(-\infty, x)$, and requires other techniques to be implemented. Are there any suggestions on proving this continuity property?! Maybe, dividing into two cases of RHS & LHS limit, along with gauging the mesh of partition?!