Layered specimens can, and often do, have discontinuities in residual stress across the material interfaces. Stress discontinuities internal to specimens are notoriously difficult to measure, yet they can have profound impact on debonding and other failures. The incremental slitting method, also known as crack compliance, is generally excellent at resolving stress profiles even when there are high gradients. However, the data reduction for slitting usually involves some smoothing, which makes it hard to resolve discontinuities. In this work, we start with a recent technique to analyze slitting data using Tikhonov regularization, and extend that technique to allow stress discontinuities at material interfaces. The new method is then demonstrated experimentally on several specimens, including metals and ceramics.