Discussing the differentiability of a multivariable function at origin.

I recently was unable to solve a Question on differntiability as follows:

Discuss the differentiability of $f$ at $(0,0)$ , where $$f(x,y) = \left\{ \begin{array}{ll} xy(x^2-y^2)/(x^2+y^2) & \quad \text{if} \space (x,y) \neq (0,0) \\ 0 & \quad \text{if} \space (x,y) = (0,0) \\ \end{array} \right. $$

I knew the necessary and sufficient condition for differentiability as:

$\lim_{Δ\rho\to0}(Δf-df)/Δ\rho=0$ , where $Δ\rho=\sqrt{x^2+y^2}$

But I have never used this condition before and am facing issues, namely computing $Δf$ (which seems to be too tiresome to compute) and what to substitute for $dx$ ad $dy$ in $df$ . Also, I tried to change to polar coordinates but couldn't do anything with $Δr$ when $r\to0$. ($r$ was $Δ\rho$ in that case)

I'm out of ideas. Would someone please help?

Solution 1:

$|xy| \leq x^2+y^2 \Longrightarrow \left|\frac{xy(x^2-y^2)}{x^2+y^2}\right| \leq |x^2-y^2| \leq x^2+y^2$. I.e., $\Delta f \leq \Delta \rho^2$, which proves that $f$ is differentiable and $df = 0$. Indeed: $\left| \frac{\Delta f - 0}{\Delta \rho}\right| < |\Delta \rho|$ and thus $\lim_{\Delta\rho\rightarrow 0} \frac{\Delta f - 0}{\Delta \rho} = 0$