= Rectangular bumps detecting nonconvergence
{title2=$B(x,y)=\sup_n\phi((x-u_n)/r_n)\phi((y-v_n)/s_n)$}
Let $(u_n,v_n)\to(0,0)$ with both coordinates nonzero, $r_n=|u_n|/4$, $s_n=|v_n|/4$, and $\phi(t)=\max(1-|t|,0)$. 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 <maximum>[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.
Back to article page
