Show that $f(x,y) = \frac{xy^2}{x^2+y^2}$ (where $(x,y) \neq0$, and $f(0,0) = 0$) is continuous but not differentiable at $(0,0)$.
I have shown continuity but am stuck on differentiability. I cannot seem to find a way in my text to show that the above is not differentiable.
Thanks
Clearly $\lim_{t\to 0^+}f(x,tx)=0$ for all $t \in \mathbf{R}$ (and also along the $y$ axis). Hence $f$ is (at least) Gateaux differentiable.
Moreover, writing $(x,y)$ in polar coordinates $(\rho \cos \theta, \rho \sin \theta)$, you get $f(x,y)=\rho\cos \theta \sin^2 \theta$. Hence $|f(x,y)| \le \rho=\sqrt{x^2+y^2}$, therefore it is continuous at $(0,0)$.
However, it is not Frechet differentiable. Indeed $$ \limsup_{(x,y)\to (0,0)}\left|\frac{f(x,y)}{\sqrt{x^2+y^2}}\right|=\limsup_{\rho \to 0}\left|\frac{\rho\cos \theta \sin^2\theta}{\rho}\right|>0. $$
