The function $xy/(x^2+y^2)$ is separately continuous but not jointly continuous

A function \(\RR^2\to\RR\) can be continuous in each variable separately without being continuous. Define \(f\colon\RR^2\to\RR\) by \[ f(x, y) = \begin{cases} \dfrac{xy}{x^2 + y^2}, & (x, y) \neq (0, 0), \\ 0, & (x, y) = (0, 0). \end{cases} \] For each \(b\in\RR\) the map \(x\mapsto f(x, b)\) is continuous on \(\RR\), and for each \(a\in\RR\) the map \(y\mapsto f(a, y)\) is continuous on \(\RR\); on the line \(y = 0\) the function vanishes identically. Along the line \(y = x\), \(f(t, t) = 1/2\) for \(t\neq 0\), so \(f(t,t)\to 1/2\neq f(0,0)\) as \(t\to 0\), and \(f\) is not continuous at the origin.