# Counterexample for the Chain rule for the Gateaux-derivative

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?

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.
• definitely I can't go any further with your hint. I started trying with an $f$ 0 everywhere but for $x \in A$. but for $g$ I don't have any idea Commented Mar 9, 2014 at 23:14
• The $y$ axis misses the whole set $A$, as does the $x$ axis. The line $x=y$ intersects $A$ for $1/2 < x < 1$ and so misses the interval $-1/2 < x < 1/2$. Commented Mar 10, 2014 at 0:23
• Try defining $g$ so that $g(t)$ is in $A$ for all $t \ne 0$. Commented Mar 10, 2014 at 0:25
• I've found another counterexample without following your hints, $h: \mathbb{R}^2 \to \mathbb{R}^2 , \ \ h(x,y) = (x,y^2)$ and $g(x,y) = \frac{y(x^2+y^2)^{3/2}}{(x^2+y^2)^2+y^2}$ if $(x,y) \neq 0$ and $g(0,0)=0$. In $(0,0)$ the chain rule doesn't hold. What was your functions instead? Commented Mar 10, 2014 at 17:49