differentiability of $(x,y)\mapsto\max\{x,y\}$

Question

At what points are the partial derivatives of the following function not defined? And how to check for the same?

$$f(x,y) = \max \{x,y\}$$

I was able to find the partial derivatives of $f$ for $x=y$ (in which case they do not exist), but I don't know how to take the limit of the partial derivative expression when $x>y$ or $x< y$ .

Answer 1

If it could help

we can observe that

$$\max(x,y)=\frac 12(x+y+|x-y|)$$