calculus - Example of a continuous and Gâteaux differentiable function that is not Fréchet differentiable.

Is there an example of a function $f:\mathbb{R}^2 \to \mathbb{R}$, with $f(0,0) = 0$, that is Gâteaux differentiable (all directional derivatives exist) and continuous at $(0,0)$, but is not Fréchet differentiable at $(0,0)$?

Edit: By Gâteaux differentiable, I use the definition that the Gâteaux derivative is not required to be a linear map, but just all the directional derivatives to exist in all directions.

Answer

Let $f \colon \mathbb R^2 \to \mathbb R$ be given by

$$f(x,y) = \begin{cases} x & x \ne 0, y = x^2, \\ 0 & \text{else}.\end{cases}$$
This function is directionally differentiable (with a linear derivative) and continuous in $(0,0)$.

With your definition of Gâteaux differentiability, you can even use any norm on $\mathbb R^2$, e.g.,

$$f(x,y) = \sqrt{x^2 + y^2}$$
or
$$f(x,y) = |x| + |y|.$$