Relation about Gateaux differentiable and differentiable Assume there is a multivariable function $f:\mathbb{R}^n\to\mathbb{R}$. I want to know the relation between Gateaux differentiable and differentiable. That is, if $f$ is differentiable, is it Gateaux differentiable? If $f$ is Gateaux differentiable, is it differentiable? Thank you.
 A: *

*$f$ is differentiable at a  point $a$ if   there is a vector $v$ such that 
$$\lim_{x\to a} \frac{|f(x)-f(a)-\langle v,x-a\rangle |}{|x-a|}=0 \tag1$$
This is also called Fréchet differentiability, to distinguish it from other kinds. 

*$f$ is  Gâteaux  differentiable at a  point $a$  for any nonzero vector $u$ the limit
$$\lim_{h\to 0} \frac{ f(a+hu)-f(a) }{h}   \tag2$$
exists. This is also called the directional derivative.

*$f$ is  Gâteaux  differentiable at a  point $a$ with linear differential if there is a vector $v$ such that for any nonzero vector $u$ we have
$$\lim_{h\to 0} \frac{ f(a+hu)-f(a) }{h}=\langle v,u\rangle  \tag2$$
The vector $v$ is called the Gâteaux differential of $f$. Normally, the words Gâteaux differential are used only in this case, when we have linearity.
It is true that (1) implies (3). The easiest way to prove this is to rewrite (1) as $f(x)=f(a)+\langle v,x-a\rangle +r(x)$ where the remainder $r(x)$ satisfies $r(x)/|x-a|\to 0$ as $x\to a$. Then plug into (2).
It is obvious  that (3) implies (2).
In spaces of dimension $n\ge 2$, (2) does not imply (3), and (3) does not imply (1). Examples are given in wikipedia article.
