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Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. Is the set of points at which $f$ is differentiable a Borel set?

The answer is "yes" for $n=1$, even for arbitrary $f$ (not assumed to be measurable).

But what happens for $n>1$? The proof for $n=1$ (refer Characterization of sets of differentiability) does not seem to generalize easily to higher dimensions.

(The motivation for asking this question came from the Rademacher's theorem, which states that any Lipschitz map $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is differentiable almost everywhere. I was wondering if the points where $f$ is not differentiable forms a Borel set. Therefore, if the solution to the original question seems obscure, please feel free to help me with partial results, especially for the Lipschitz case.)