TITLE: Real Analysis Question from Terence Tao's Book
QUESTION [6 upvotes]: I was having trouble with what I believe to be a rather easy problem from section six of Terence Tao's Introduction to measure theory.  Help will be appreciated.
If $F$ is everywhere differentiable (and thus $F$ is continuous), show $F'$ is measurable.

REPLY [8 votes]: Define:
$$
g_n(x) = \frac{F(x + 1/n) - F(x)}{1/n}
$$
This is a sequence of measurable functions. The pointwise limit as $n \to \infty$ is $F'$. Since the pointwise limit of a sequence of measurable functions is measurable, it follows that $F'$ is measurable.