A function $\varphi[a,b] \to \mathbb{R}$ is said to be singular if
- $\varphi \in C[a,b]$ (i.e., $\varphi$ is continuous on $[a,b]$),
- $\varphi'(x)$ exists a.e. in $[a,b]$,
- $\varphi'(x)=0$ a.e. in $[a,b]$.
Let $f$ be continuous on $[a,b]$ and of bounded variation on $[a,b]$.
Prove that there is an absolutely continuous function
$F: [a,b] \to \mathbb{R}$ and a singular function
$\varphi: [a,b] \to \mathbb{R}$ such that $f = F + \varphi$.
Best Answer
Any function of bounded variation is a difference of two monotone increasing functions. Just note that linear combinations of absolutely continuous functions are absolutely continuous and linear combinations of singular continuous functions are singular.