Prove or Disprove that if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ both exist, then f is continuous at $(x_0,y_0)$

Question

I have to either prove or disprove the fact that if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ both exist, then f is continuous at $(x_0,y_0)$.

What I thought:

I thought that the best way to approach this is to use a function that does what we want to prove or disprove. So my attempt to finding a function is:

$$f(x,y)=\left\{\begin{array}{cl} \frac{xy}{x^2+y^2} & \text{if } (x,y) \neq (0,0)\\ 0 & \text{ } \text{otherwise}\end{array}\right.$$

Is this correct to use? What do I do next?