Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is a function. And $f_{xx}$ and $f_{yy}$ exist and continuous. Is $f$ a twice differenced function?

Asked May 25 '19 at 18:34

Active May 25 '19 at 19:49

Viewed 148 times

Obviously, it is true if $\delta>0$ imply $2f(x+\delta, y+\delta)-f(x+2\delta, y)-f(x, y+2\delta)=o(\delta^2)$. But this result I can't prove too.

edited May 25 '19 at 19:49

cmk

asked May 25 '19 at 18:34

Andrey Komisarov

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I think it is anouth to proof that f_xy exist. – Andrey Komisarov May 25 '19 at 18:35
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It is true. $f$ is differentiable if the partial derivatives exist and are continuous – Syd Amerikaner May 25 '19 at 22:56
@SydAmerikaner But how can I prove this? – Andrey Komisarov May 26 '19 at 00:21
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see for example here https://math.stackexchange.com/questions/1007709/proof-that-continuous-partial-derivatives-implies-differentiability – Syd Amerikaner May 27 '19 at 18:50
@SydAmerikaner I don't see how the result at that link applies here. – David C. Ullrich Aug 22 '19 at 17:39
If all partial derivatives exist and are continuous. This is the result in the link. So if one can prove that $f_{xy}$ and $f_{yx}$ exist and are continuous, so is the derivative of $f$ differentiable, i.e. the second derivative of $f$ exists. – Syd Amerikaner Aug 22 '19 at 17:56
@SydAmerikaner But how do we show $f_{xy}$ even exists? – David C. Ullrich Aug 22 '19 at 17:59

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