In our analysis 2 lecture, we learned that if a function is Gâteaux differentiable, the function might not be Fréchet differentiable. One example we saw was that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by

$$ f(x, y) = \begin{cases} \frac{x^2y}{x^4 + y^2}, x, y \ne 0 \\ 0, x = y = 0 \end{cases}$$

We know that $G(a, b) = a^2/b$ if $b \ne 0$ and $G$ is not linear everywhere so $f$ is not Fréchet differentiable. But does this generally mean if $G$ is linear everywhere then $f$ is Fréchet differentiable?



No, not necessary. See below from the Wikipedia page enter image description here


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