a mesurable function and bounded a.e is of bounded variation?

Question

Let $F:[0,1]\rightarrow \mathbb{R}$ be a measurable function, bounded a.e

Does this imply that $F$ is of bounded variation on $[0,1]$?

Answer 1

No, let $F$ be the indicator of the rational numbers. Then $F$ is bounded everywhere, measurable, and infinite variation.

Answer 2

Not even true if we assume in addition that $F$ is continuous: \begin{align} F(x) &= \sin\frac{1}{x},\quad 0<x\le 1, \\ F(0) &= 0 \end{align}