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 Pᵖ/Qq 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 Łojasiewicz.