Partial derivative and limit

Question

The function $ f(x,y)=0$ $if$ $x,y≠0$ $and$ $ f(x,y)=1$ $if$ $ x,y=0$ is given. I have to prove that partial derivative of $x$ and partial derivative of $y$ exist at the beginning of the axes and find them.I am confused about how we find the derivatives.

Answer 1

Partial derivatives exist for any $ (a,b)\neq(0,0)$, let proceed by the definition

$$f_x=\lim_{h\to0}\frac{f(a+h,b)-f(a,b)}{h}$$

$$f_x=\lim_{h\to0}\frac{f(a,b+h)-f(a,b)}{h}$$