Continuous function: Borel and/or Lebesgue measurable? If $f\colon [0,1]\to\Bbb R$ is a continuous function and differentiable on (0,1), then which of the following is correct?


*

*$f'$ is Borel and Lebesgue measurable.

*$f'$ is Borel measurable and is not Lebesgue measurable.

*$f'$ is not Borel measurable and is not Lebesgue measurable.

*$f'$ is not Borel measurable and is Lebesgue measurable.


This might help.
$$f'(x) = \lim_{n\to\infty} n(f(x + 1/n) - f(x)).$$
