relation between total derivative and directional derivative, Gateaux and Frechet derivative

Question

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!

Answer 1

No, the converse is not true (unless $n=m=1$, where the equivalence is trivial), and the typical example is that of a non-linear homogeneous function, like $f:\Bbb{R}^2\to\Bbb{R}$ defined by \begin{align} f(x,y):= \begin{cases} \frac{x^3}{x^2+y^2} & \text{if $(x,y)\neq (0,0)$}\\ 0 & \text{if $(x,y)=(0,0)$} \end{cases} \end{align}A sufficient condition for Frechet differentiability is that we need all the directional derivative to be continuous: for every $h$ lying in a spanning set of $\Bbb{R}^n$, we need the function $x\mapsto (D_hf)(x):=\frac{d}{dt}\bigg|_{t=0}f(x+th)$ to be continuous.

I would recommend you read through Loomis and Sternberg's Advanced Calculus section 3.7 and 3.8 (page 148 if you only want the explanation of the counter example, but I recommend you read both these sections completely; perhaps after reading 3.6 to get acquainted with the notation).