Derivatives are Borel functions
It is known [\textit{H. Federer}, Geometric measure theory. Berlin etc.: Springer-Verlag (1969; Zbl 0176.00801), p. 211] that the differential of any continuous mapping of a finite dimensional normed space into a Banach space is a Borel function, defined on a Borel set. The author extends this result to noncontinuous functions with separable image. He proves the following Theorem: Let \(X\) be a finite dimensional normed space, \(Y\) a separable Banach space, and let \(f: X\to Y\) be an arbitrary function. Then \((Df)^{-1}(F)\) is in \({\mathcal F}_{\sigma \delta}\) whenever \(F\) is a closed subset of \(\mathcal B(X,Y)\), the space of all continuous linear transformations of \(X\) into \(Y\). [For the notation see Federer's book cited above.]
scientific article; zbMATH DE number 4006572