# Composition of almost everywhere differentiable functions

Let $f: \mathbb R^n \rightarrow \mathbb R^m$ and $g: \mathbb R^m \rightarrow \mathbb R^k$ be almost everywhere differentiable functions (with respect to Lebesgue measure). Is it true that $g \circ f$ is differentiable almost everywhere?

In the special case $m=n=k$ is it true?

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## 1 Answer

No, $g \circ f$ need not even be continuous anywhere. The trouble is that the image of $f$ could be concentrated on a "small" set where $g$ is bad. For example, let $f: {\mathbb R} \mapsto {\mathbb R}^2$ and $g: {\mathbb R}^2 \mapsto {\mathbb R}$ with \eqalign{f(x) &= (x,0)\cr g(x,y) &= \cases{0 & if y \ne 0\cr \chi_{\mathbb Q}(x) & if y = 0\cr}} where $\chi_{\mathbb Q}$ is the indicator function of the rationals. Then $g \circ f = \chi_{\mathbb Q}$ is nowhere continuous.

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