Jacobian matrix and Gateaux derivative Let $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ be a function which is Gâteaux differentiable and let $J_f\in\mathbb{R}^{M\times N}$ be its Jacobian matrix.
Is it true that the Gâteaux derivative of $f$ along a direction $v\in\mathbb{R}^N$ is equal to the matrix-vector product $J_f \cdot v$?

The answer in general is no. The classic counter example is given by the function
$$
f(x,y)=\left\{
\begin{array}{cc}
\frac{x^2y}{x^4+y^2} & (x,y)\neq (0,0),\\
0 & (x,y)=(0,0)
\end{array}
\right.
$$
You can check that at $(0,0)$ this function has all the directional derivatives. However, it is neither differentiable, nor continuous. In particular, the partial derivatives are both zero, but along all other directions its derivative is nonzero.
