Help with differentiable function $f:\mathbb{R^2} \to \mathbb{R}$ I need help with a question that appeared in my test. 
True or False: 
Let $f$ be a function $f:\mathbb{R^2} \to \mathbb{R}$  not differentiable at (0,0), then $f^2$ is not differentiable at (0,0).
I answered False. I gave an example of $\sqrt{x+y-1}$ which isn't defined at (0,0) let alone differentiable, yet $x+y-1$ does. But my professor wrote on my test that $\sqrt{x+y-1}$ is differentiable at (0,0). 
who among us is incorrect?  

The function you specified is not defined on $\mathbb{R}^2$.
Try $f(x,y) = \sqrt{|xy|}$. Then $f$ is not differentiable at $(0,0)$, but $f(x,y)^2 = |xy|$ is differentiable at $(0,0)$ (since $|xy| \le \frac{1}{2} (x^2+y^2)$).
