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Does there exist a function $f:\mathbb{R}^n\to \mathbb{R}^m$ with the property:

$\exists x_0\in\mathbb{R}^n$ such that $\frac{\partial f_i}{\partial x_j}(x_0)$ exists for all $i,j$, but $f$ is discontinuous at $x_0$.

I konw that all partial derivative exist does not imply total differentiability. But what about continuity? I am curious about this.

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up vote 2 down vote accepted

Define $f: \mathbb{R^2} \to \mathbb{R}$ by $f(x) = \begin{cases} 0, & x_1 = 0 \mbox{ or } x_2 = 0 \\ 1, & \text{otherwise}\end{cases}$. Then the partials exist at $x_0=(0,0)$, but the function is not continuous at $x_0$.

The point being that the partials, in general, only provide information about the behavior of the function along lines parallel to the axes.

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