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**heathrowjohnny** Lets say we have the scalar field $\displaystyle f(x,y) = \begin{cases} \frac{xy}{\sqrt{x^2+y^2}} \ \text{if} \ (x,y) \neq (0,0) \\ 0 \ \ \ \ \ \ \ \ \ \text{if} \ (x,y) = (0,0) \end{cases} $

So by inspection, it seems that $\displaystyle f(x,y) $ is continuous at $\displaystyle (0,0) $. Now the partial derivatives of $\displaystyle f $ are not continuous at $\displaystyle (0,0) $. So $\displaystyle f $ is not differentiable at $\displaystyle (0,0) $. So this is not Fréchet differentiable? But could it be Gâteaux differentiable?