On Convex Functions

Today I came across a result for AC functions.

Theorem. If $f :I\to \R$ is absolutely continuous with $f'\in BV(I)$, then $f$ is representable as a difference of two convex functions.

We need this result for generalised Ito's Lemma which works on convex functions (with an indirect route: consideration of local time process). This result seems nonstandard, and I have paid a few hours finding standard results on convex functions, the following turns out to be what I want:

Lemma (Thm 14.14 of J. Yeh's Real analysis). Let $f$ be a real-valued function an open interval $I$ in $\R$. Suppose that

1) $f$ is AC on any closed subinterval of $I$; and
2) $f'$ is increasing on the subset, $A$, of $I$ on which $f'$ exists and $m(A)=m(I)$.

Then $f$ is a convex function on $I$.

Having the lemma we can prove the Theorem immediately.

Proof. Since $f$ is $AC$, fix $a\in I$ and for any $x\in I$ we have \[f(x)=f(a)+\int_a^xf'(s)\,ds.\] Now we can decompose $f'$ into a difference of two nondecreasing functions since $f'\in BV$, call them $H, K$, i.e., $f'=H-K$. As a result, \[
f(x) = f(a) +\int_a^x H\,ds - \int_a^x K\,ds,
\] finally we denote $h = \int_a^x H\,ds$ and $k = \int_a^x K\,ds$. Then $h,k$ are AC on any closed subinterval of $I$, moreover, $h' = H,k'=K$ a.e. and they are increasing, therefore we can apply the lemma to conclude that $h$ and $k$ are convex.$\qed$

There are many interesting and fundamental facts for convex functions that are not mentioned in UG curriculum of UST and I really suggesting reading them all, they are too standard to miss.

Still I want to record an important fact of convex functions: $f'_-(x)$ is increasing (of course) and always left continuous if $f$ is convex, this makes if possible to define Stieljes measure by using $f'_-$.