 Solution

Let $V(t)$ be the total variation of $f$ on $[0,t]$. The Jordan decomposition of a function of bounded variation isFor $s<t$, the inequality $V(t)-V(s)\ge |f(t)-f(s)|$ shows that both increments are nonnegative, so $g,h$ are nondecreasing and $f=g-h$. Right-continuity of the finite variation function $f$ implies right-continuity of $V$, and hence of $g,h$.