Constructing Discontinuous but Locally Bounded Rational Functions using Łojasiewicz Inequalities

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 $P^p\mathrm{/}{Q}^q$ 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.