Separate continuity

A function of several variables is separately continuous when it is continuous in each variable with the others fixed. This is weaker than joint continuity in the product topology. Functions with narrow rectangular supports provide useful examples distinguishing convergence in topologies generated by separate continuity from Euclidean convergence.

Let $(u_n,v_n)\to (0,0)$ with both coordinates nonzero, $r_n=|u_n|/4$, $s_n=|v_n|/4$, and $\varphi (t)=\max (1-|t|,0)$. The displayed function equals one at every $(u_n,v_n)$ and zero on both coordinate axes. Each coordinate line off the corresponding axis meets only finitely many supports, so its restriction is a finite maximum of continuous functions. On the axes it is zero. Hence $B$ is separately continuous and disproves convergence of this sequence in any topology making every separately continuous function continuous. Isotropic radial bumps need not have this property when one coordinate approaches zero much faster than the other.