Writing a continuous function of bounded variation as the sum of a singular function and an absolutely continuous function.

Question

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 $f = F + \varphi$.

It is true for a monotone increasing function (see here). – J. Bishop Jan 14 '20 at 23:09 — J. Bishop, Jan 14 '20 at 23:09

score 1 · Accepted Answer · answered Jan 14 '20 at 23:29

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.

Writing a continuous function of bounded variation as the sum of a singular function and an absolutely continuous function.

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