We discuss projection based Pad ́e-Jacobi approximants in general and present in particular an exact rational approximation to the Sign function. This serves as vehicle to analyze the behavior of Pad ́e-Jacobi approximants for discontinuous functions. The analysis shows that the Pad ́e-Jacobi approximant is superior in several ways to classic polynomial approximations of discontinuous functions, provided the parameters in the approximations are chosen carefully. Guidelines for this is obtained through the analysis.