differentiability of $|xy|$ at $(0,0)$

Question

Let, $f(x,y)=|xy|$, Show that $f$ is differentiable at $(0,0)$

I started with by trying to prove $\lim_{h\to 0,k\to 0}\dfrac{f(0+h,0+k)-f(0,0)}{hk}$ exists. Then, $\lim_{h\to 0,k\to 0}\dfrac{|hk|}{hk}=\lim_{h\to 0,k\to 0}\dfrac{|h||k|}{hk}=\lim_{h\to 0}\dfrac{|h|}{h}\lim_{k\to 0}\dfrac{|k|}{k}$

Can I do this? Because, if I do this, $f$ may not be differentiable, but I have to show the converse. Thanks, for any help!

Answer 1

Your approach is not correct.

We have $ \frac{f(t,0)-f(0,0)}{t}=0 \to 0$ as $t \to 0$. Hence $f_x(0,0)$ exists and $=0$. The same arguments show that $f_y(0,0)$ exists and $=0$.

Now consider

$Q(x,y):= \frac{f(x,y)-f(0,0)-xf_x(0,0)-yf_y(0,0)}{\sqrt{x^2+y^2}}$.

Then we have: $f$ is differentiable at $(0,0) \iff Q(x,y) \to 0$ as $(x.y) \to (0,0)$.

It is your turn to show that $Q(x,y) \to 0$ as $(x.y) \to (0,0)$.