Let $f\colon\mathbb{R}^{n}\to\mathbb{R}$ be continuous everywhere. If we know that in one single point $x_0$ all the partial derivatives exist and are linear, $\partial_vf(x_0)=L(v)$, does this imply that $f$ is totally differentiable in $x_0$?

In a previous question, user Etienne gave an example that showed that the continuity of $f$ everywhere rather than just in $x_0$ is necessary.

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    $\begingroup$ Do you mean directional derivatives? $\endgroup$ – dfeuer Jul 20 '13 at 0:20

The answer is still "no". In fact, we can smooth out Etienne's example from the previous question.

Let $B\colon\mathbb{R}\to[0,1]$ be a smooth bump function supported on $[0,2]$ such that $B(1) = 1$. Now let $f(x,y)=xB(y/x^2)$.


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