Ways to represent functions of bounded variation as the difference of two monotone functions

Question

I learnt (it is called Jordan's representation theorem) that every function $f : [a,b] \to \mathbb R$ with bounded variation could be represented as the difference $f = g - h$ of two monotone functions. In the proof, two monotonically increasing functions $g$ and $h$ are constructed, so every function with bounded variation could be represented as the difference of two monotonically increasing functions. Now I am asking, is there is any function with bounded variation which could not represented as the difference of two function, i) one monotonically increasing, the other monotonically decreasing, or ii) such that both are decreasing?

Answer 1

The answers are 1) yes and 2) no. If $g$ is increasing and $h$ is decreasing (or vise-versa), then $g-h$ is monotone. A non-monotone BV function cannot be represented this way. On the other hand, if $f$ is $BV$ and is represented as $f = g - h$ where $f$ and $g$ are increasing, you can also write $f = (-h) - (-g)$ so that $f$ is a difference of two decreasing functions.