Currently, I am following a portion of the text Methods of Nonlinear Analysis. For normed real vector spaces $(X,\|\cdot\|_X),(Y,\|\cdot\|_Y)$ and $a \in X$, consider the following definitions from this text.
Directional Derivative: Let $h \in X$. If the limit $$\lim_{t \to 0;\ t \in \mathbb{R}} \frac{f(a+th)-f(a)}{t}$$ exists in $Y$, then the value of the limit is called the derivative of $f$ at $a$ in the direction $h$ and is denoted $\delta f(a;h)$.
Gâteaux Derivative: Suppose $\delta f(a;h)$ exists for all $h \in X$. If the function $Df(a) : X \to Y$ where $h \mapsto \delta f(a;h)$ is continuous and linear, then $Df(a)$ is called the Gâteaux derivative of $f$ at $a$.
Fréchet Derivative: If there exists a continuous linear transformation $A : X \to Y$ such that $$\lim_{\|h\|_X \to 0} \frac{\|f(a+h)-f(a)-Ah\|_Y}{\|h\|_X}$$ then we call $A$ the Fréchet derivative of $f$ at $a$ and denote it $f'(a)$.
Question: I am struggling to find an example of $X,Y,a,f$ such that the Gâteaux derivative of $f$ exists at $a$ but the Fréchet derivative of $f$ does not exist at $a$. Can anyone help provide such an example? Please note that the Gâteaux derivative must be linear and continuous. Since $f$ having a Fréchet derivative at $a$ implies the continuity of $f$ at $a$, I imagine there exists an example with $X = \mathbb{R^2}, Y = \mathbb{R}, a = (0,0)$ such that $Df(0,0)$ exists but $f$ is not continuous at $(0,0)$.
Among many other online sources, I have read the following stack exchange posts but I believe that each asks a slightly different question or uses a slightly different definition of the Gâteaux derivative:
Example of a continuous and Gâteaux differentiable function that is not Fréchet differentiable.
Question about discontinuous function with directional derivatives at a points
After more thinking and searching, I have found an example from this Wikipedia page which I have verified answers my question.
Suppose $X = \mathbb{R}^2$, $Y = \mathbb{R}$, and $a = (0,0)$. Moreover, for $(x_1,x_2) \in \mathbb{R}^2$, let $$ f(x_1,x_2) := \begin{cases} 0 & \text{if } (x_1,x_2) = (0,0) \\ \frac{x_1^3x_2}{x_1^6+x_2^2} & \text{otherwise.} \end{cases} $$ Then, one computes by definition that the Gâteaux derivative of $f$ at $(0,0)$ exists and is the zero linear transformation. Yet, $f$ is not continuous at $(0,0)$ since $f(x,x^3) = 1/2$ for all $x \in \mathbb{R} \setminus \{0\}$. If $f$ were Fréchet differentiable at $(0,0)$, it would have to be continuous at $(0,0)$. Altogether, then, we have found a function $f$ such that the (linear and continuous) Gâteaux derivative exists at $(0,0)$ but the Fréchet derivative does not exist at $(0,0)$.