Let $f(x)=\sin(x)$ on $x \in[0,2\pi]$. Find two increasing functions $h$ and $g$ such that $f=g-h$ on $x \in [0, 2\pi]$.
Finding the explicit example is where I'm stuck. Since this is a bounded function of finite tototal variation I know an explicit $h$ and $g$ exists. I just don't know what it is.