A function whose partial derivatives exist at a point but is not continuous

You may also consider this function: $$ f(x,y) = \left\{ \begin{array}{cc} \displaystyle\frac{xy}{x^2+y^2} & (x, y)\neq (0,0) \\ 0 & \text{otherwise.} \end{array}\right. $$ If you check, you see that this function is not continuous at $(0,0)$ but both partials exist and equal zero, that is, $f_x(0,0)=f_y(0,0)=0$.

On $\mathbb R^2,$ define $f=1$ except on the axes, where we define $f=0.$ Then $f_x(0,0)= f_y(0,0) = 0,$ but $f$ is not continuous at $(0,0).$

Hint: this is the graph of $f(x,y)=\frac{(2x^2-3y^2)^2}{(x^2+y^2)^2}$ over $[-1,1]^2$:

To consider functions which are constant over lines through the origin is a very good idea.