If $f : I\to \mathbb{R}$ is differentiable almost everywhere, where $I$ is an open interval, show that $f'$ is Lebesgue measurable.
I know that pointwise limits of measurable functions are measurable and that if $f$ is differentiable everywhere, $f'$ is measurable (for instance, I can use a technique similar to this post). I also know that $f'$ must agree with its restriction to $I\backslash E$ almost everywhere, where $E$ is the set of points on which $f$ isn't differentiable.
My issue is that $f'$ may not even be defined sometimes, so why can't we just restrict $f'$ to $I\backslash E$ and conclude it's measurable because it's only defined on $I\backslash E$?