Is the set of differentiable points of a monotonic function Borel-measurable?

Asked Apr 15 '16 at 17:48

Active Apr 15 '16 at 17:48

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Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a monotonically increasing function.

Then, it has a countable discontinuity and is differentiable almost everywhere with respect to the Lebesgue measure.

Define $D=\{x: f \text{ is differentiable at } x\}$.

I know that $m(D^c)=0$ and $D$ is Lebesgue-measurable. However, I'm curious whether $D$ is Borel-measurable. Is there an example such that $D$ is not Borel-measurable?

asked Apr 15 '16 at 17:48

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In fact, the set of points of finite differentiability of ANY function (even a non-Lebesgue measurable function) from the reals to the reals is a $F_{\sigma \delta}$ set, which is Borel measurable in a fairly simple way. See my answer at Characterization of sets of differentiability. See also the answers at Continuous functions are differentiable on a measurable set?. – Dave L. Renfro Apr 15 '16 at 18:04

Is the set of differentiable points of a monotonic function Borel-measurable?

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