One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:

$$\label{p}\tag{P} f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,b). $$

We know that a convex function $g:(a,b)\to \mathbb{R}$ is locally Lipschitz and its second derivative $g''$ exists a.e. $x\in (a,b)$.

Can we find a increasing convex function $g$, such that $g''$ satisfies the property \ref{p}?

If the answer is negative

What can we say about the set of intervals $I$ for which $g''\notin L^1(I)$?

Remark: I've asked a similar question on Math.stack with no answer.