A function with partial derivatives but not continuous

Question

I have a function f(x,y) which is 1 at the origin and 0 for any other point. This function isn't continuous because the limit doesn't exist at (0,0). Could you show its partial derivatives exist. (not homework)

Answer 1

For example:

$$\frac{\partial f}{\partial x}(0,0):=\lim_{x\to0}\frac{f(x,0)-f(0,0)}x=\lim_{x\to0}\frac{0-1}x$$

so nop: the partial derivatives don't exist at the origin. They do at any other point, though.