So I have the function $$f(x,y) =\begin{cases} xy\frac{x^2-y^2}{x^2+y^2} & (x,y) \neq (0,0),\\ 0 & (x,y)=(0,0). \end{cases}$$ I need to tell if the function is Frechet differentiable and why.
I tried to find out something on the internet but I couldn't manage to solve it.
Can you please help me with this?
Since $f_x(0,0)=fY(0,0)=0$ we have to investigate if
$$g(x,y):= \frac{f(x,y)}{\sqrt{x^2+y^2}} \to 0$$
as $(x,y) \to (0,0).$
An easy eatimation gives
$$|g(x,y)| \le \frac{|xy|}{\sqrt{x^2+y^2}}$$
Since $2|xy| \le x^2+y^2$, we derive
$$|g(x,y) \le \frac{1}{2}\sqrt{x^2+y^2}.$$
Conclusion ?
