Consider a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$.
Given the total derivative $df(x,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}^m$, we can get the directional derivative $df(x,h)$ for every direction $h\in\mathbb{R}^n$.
However, if the directional derivative exists for every direction, does the total differential also exist, i.e., if we simply put all directional derivative together, do we get the total derivative? My guess is no. I think we need $f$ to be continuously differentiable.
I am asking this question because I try to figure out the difference between Gateaux derivative and Frechet derivative. I realize that the Gateaux derivative is the generalization of directional derivative and the Frechet derivative is the generalization of the total derivative, i.e., their definitions are the same in $\mathbb{R}^n$. Any explanation about the difference of Gateaux and Frechet will be very helpful! Thank you!