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Does there exist a function $f$ on an open set $G\subset\mathbb{R}^2$ , which $f_x$ and $f_y$ exist everywhere but $f$ is nowhere differentiable in $G$?

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If we assume that $f$ is continuous then no. In that case $f_x$ and $f_y$ are both in Baire class $1$; hence they are both continuous on a comeager set, so there exists a point where they are both continuous.

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  • $\begingroup$ And what if $f$ has some discontinuous points? $\endgroup$ – Antimonius Jun 10 '18 at 1:07

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