Differentiable function has measurable derivative?
Let $f:[0,T] \to \mathbb{R}$ be a differentiable function. Is it true that $f'$ is measurable?
If so, is this also true if $f$ is differentiable almost everywhere?
Sorry for lack of effort but I don't have any clue about the answer.
Here's a key observation: Assume that $f$ is differentiable and define
$$f_n(x) = \left\{ \begin{array}{cl} \frac{f(x+1/n)-f(x)}{1/n} & \text{if } 0 \leq x+1/n \leq T \\ 0 & \text{else.} \end{array}\right.$$
Then, each $f_n$ is measurable and $f_n \rightarrow f'$.