THEOREM $3.1$: If $f_{xy}(x, y)$ and $f_{yx}(x, y)$ are continuous on an open set containing $(a, b)$, then $f_{xy}(a, b) = f_{yx}(a, b)$.
My question is: Could $F_{xy}$ be continuous while $F_{yx}$ is not? Is there example for this case ?
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Mohamed Mostafa
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It is certainly possible for $f_{xy}$ to be continuous while $f_{yx}$ does not exist. This is one sense in which "$f_{yx}$ is continuous" could be false. – Erick Wong Jul 17 '16 at 16:01
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Could you please give an example of this case? – Mohamed Mostafa Jul 17 '16 at 23:34
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Just choose a sufficiently non-nice function which depends only on $y$. Then the $y$-derivative won't be defined, so $f_{yx}$ does not exist, but $f_{x} = 0$ everywhere so $f_{xy} = 0$. – Erick Wong Jul 18 '16 at 01:16