Show that f is not differentiable at (0,0)

Question

Let $f:\mathbb{R}^2\to\mathbb{R}$ be defined by $f(x,y)=\sqrt{|xy|}$. Show that $f$ is not differentiable at $(0,0)$.

If you could start me out on how to show this, that would help a lot.

Answer 1

Let us approach along the direction $(x,x)$ (i.e. the line $y=x$) for the difference quotient. Then we note that on this we have $f(x,x)=\sqrt{x^2}=|x|$ which we know is not differentiable at $x=0$, hence the function fails to be differentiable.