Assumption for separate continuity implies joint continuity
Solution 1:
Unfortunately,
$$f(x,y) = \begin{cases}\qquad 0 &, x = y = 0\\ \dfrac{xy}{x^2+y^2} &, x^2+y^2 > 0\end{cases}$$
is separately continuous on $[-1,1]^2$.
Unfortunately,
$$f(x,y) = \begin{cases}\qquad 0 &, x = y = 0\\ \dfrac{xy}{x^2+y^2} &, x^2+y^2 > 0\end{cases}$$
is separately continuous on $[-1,1]^2$.