Locally Lipschitz and Gâteaux Derivative if and only if Frechet Derivative Consider $f$ locally Lipschitz. So $f$ is Gâteaux Derivative if and only if $f$ is Frechet Derivative. 
PS.: the converse is trivial.

Seems to me the result is false.
Say $S$ is the unit circle in $\mathbb R^2$. Say $\phi:S\to\mathbb R$ is any smooth function with $\phi(-x)=\phi(x)$. There's a function $f:\mathbb R^2\to\mathbb R$ such that 


*

*$f(x)=\phi(x)\quad(x\in S),$

*$f(tx)=tf(x)\quad(x\in\mathbb R^2,t\in\mathbb R).$
This seems like a counterexample to me, unless it just happens that $\phi$ is the restriction to $S$ of a linear functional. What am I missing?
