Define $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ by $$ f(x,y)= \begin{cases} \frac{xy}{x^2+y^2} &\text{if}\, (x,y)\neq(0,0)\\ 0 &\text{if}\, (x,y)=(0,0) \end{cases} $$
I have found that for $(x,y)\neq(0,0)$ $$\frac{\partial f}{\partial x}=\frac{y(y^2-x^2)}{(x^2+y^2)^2}$$ and $$\frac{\partial f}{\partial y}=\frac{x(x^2-y^2)}{(x^2+y^2)^2}$$
I don't know how to show that these are not continuous functions. I know there exists similar questions, but there is no duplicate of this question since I am asking about the continuity of the partial derivatives not of $f(x,y)$
By direct calculation, you'll get $f_x(0, 0) = 0$ and $f_y(0, 0) = 0$. So you can check whether $\lim_{(x, y)\to(0, 0)}f_x(x, y) = 0$ or not, and the same for $f_y$. Can you find suitable path s.t $f_x$ does not converge to $0$?