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I'm struggling to prove that $ \varphi\left ( a,b \right )= \frac{x^{3}y}{x^{4}+y^{2}} $ is Gâteaux differentiable at $0$, but not Fréchet differentiable. In other examples that I studied, proving that the function wasn't continuous at $(0,0)$ and that it was Gâteaux differentiable at the same point was sufficient, but I am having trouble to prove the continuity of the function above in $(0,0)$

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Continuity at $(0,0)$ is easy: $2x^{2} |y| \leq x^{4}+y^{2}$ so $|\phi (a,b) | \leq \frac 1 2 |x| \to 0$.

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