Constructing discontinuous but locally bounded rational functions
using Lojasiewicz inequalities

Adam Coffman, Yifei Pan

article abstract

For real multivariate polynomials P and Q both vanishing at a point,
if the zero set of Q is contained in the zero set of P, then there
exists a rational function of the form P^p/Q^q which is locally
bounded and such that its extension that vanishes on the zero set of Q
is discontinuous.  The proof uses inequalities of Lojasiewicz.