Counterexample for the Chain rule for the Gateaux-derivative

Question

I'm reading the book of Drabek, Milota - Methods of Nonlinear Analysis, and at page 121, they state:

but I can't manage to find such counterexample. For clarity the Gateaux derivative is defined in this way:

I need some kind of hints about how to build such counterexample because I'm like going nowhere with my trials. According to me $f$ and $g$ can't be continuous, otherwise G-derivative would be Frechét-derivative and for this kind of derivative the chain rule holds. It is sufficient requiring that only one function is non-continuos?

Answer 1

Hint: define $f: {\mathbb R}^2 \to {\mathbb R}$ such that $f(x,y) = 0$ unless $ x^2 < y < 2 x^2$. Note that the intersection of any line through the origin with the exceptional set $A = \{(x,y): x^2 < y < 2 x^2\}$ misses some interval around the origin, so what $f$ does in $A$ does not affect the Gâteaux derivative at the origin.