If $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ has partial derivatives everywhere and they are continuous in the point $ (0,0) $ is $f$ differentiable?

Asked Sep 18 '23 at 07:54

Active Sep 18 '23 at 07:54

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I stumbled upon this problem:

Given a function $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ has partial derivatives everywhere and they are continuous in the point $ (0,0) $ is $f$ differentiable in $ (0,0) $?

We had a proof in lecture that states, if the partial derivatives are continuous everywhere that the function needs to be differentiable in every point. I guess the proof would also work if the partial derivatives are only continuous in a single point that the function is certainly differentiable in that point.

Any clarification would be very much appreciated.

asked Sep 18 '23 at 07:54

user007

1

Hint: a function differentiable at origin must take the form $f(r,\theta)=c+r(a\cos\theta+b\sin\theta)+o(r)$ in polar coordinates. – user10354138 Sep 18 '23 at 08:50
So it is correct since $ r(a cos( \theta ) +b sin( \theta )) $ goes to $ 0 $ and so is $ o(r) $ as $ r \rightarrow 0$ ? – user007 Sep 18 '23 at 08:58
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It's true also under slightly weaker conditions. See this answer: https://math.stackexchange.com/a/53711/1242. – Hans Lundmark Sep 18 '23 at 09:29
@HansLundmark thanks for the answer – user007 Sep 18 '23 at 11:09

If $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ has partial derivatives everywhere and they are continuous in the point $ (0,0) $ is $f$ differentiable?

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