Variably Scaled Kernels (VSKs) replace the global shape parameter with a scaling function, which maps data to an augmented feature space. This construction preserves positive definiteness while increasing separation distances and potentially improving stability. We extend VSKs to discontinuous scaling functions (VSDKs) for approximating functions with jumps, effectively mitigating the Gibbs phenomenon. We establish that VSDKs are limits of continuous VSKs and prove error bounds in piecewise Sobolev spaces, which allow us to show convergence rates that depend only on smoothness within subdomains, not on jump discontinuities.