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Let $f : \mathbb R^n \rightarrow \mathbb R^m$ and $a \in \mathbb R^n$such that $\forall i \in [1,n], \large \frac{\partial f}{\partial x_i}(a)$ exists.

Is $f$ Frechet differentiable ?

I'd say no, but I don't know any counter-examples...

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The answer becomes positive if the partial derivatives exist and are continuous functions. – Siminore May 11 '14 at 9:53
up vote 4 down vote accepted

Let $n = 2$, and $m = 1$, and consider

$$f(x) = \begin{cases} \frac{xy}{x^2+y^2} &, (x,y) \neq (0,0)\\ \quad 0 &,(x,y) = (0,0).\end{cases}$$

Then $f$ isn't even continuous at $(0,0)$, but the partial derivatives exist. (The partial derivatives exist on all of $\mathbb{R}^2$. $f$ is Fréchet-differentiable everywhere except at $(0,0)$.)

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