Let \(F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) be defined by the equation
\[ F(x \times y) = \begin{cases} xy/(x^2 + y^2) & \text{if } x \times y \neq 0 \times 0, \\ 0 & \text{if } x \times y = 0 \times 0. \end{cases} \]
Show that \(F\) is continuous in each variable separately.
Compute the function \(g: \mathbb{R} \to \mathbb{R}\) defined by \(g(x) = F(x \times x)\).
Show that \(F\) is not continuous.
Solution
Fix \(x_0\in\mathbb R\). If \(x_0=0\), then \[ y\longmapsto F(0,y) \] is identically zero, hence continuous. If \(x_0\ne0\), then \[ y\longmapsto \frac{x_0y}{x_0^2+y^2} \] is an ordinary continuous rational function, since the denominator never vanishes. Thus \(F\) is continuous in the second variable with the first fixed. The same argument, by symmetry, proves continuity in the first variable with the second fixed.
Along the diagonal, \[ g(x)=F(x,x)= \begin{cases} \frac12,&x\ne0,\\ 0,&x=0. \end{cases} \]
The diagonal map \(d:\mathbb R\to\mathbb R^2\), \(d(x)=(x,x)\), is continuous. If \(F\) were continuous, then \(g=F\circ d\) would be continuous. But \(g(x)=1/2\) for all \(x\ne0\) and \(g(0)=0\), so \(g\) is discontinuous at \(0\). Hence \(F\) is not continuous.