On the pointwise existence of Cauchy $\rm{P.V.}$ integrals

The Riemann-Hilbert (RH) approach, whose origins can be traced back to Riemann's PhD thesis, is well known to be far-reaching. It provides a general framework for expressing solutions of integrable problems such as ODEs or PDEs. Its generalization concerning monodromy groups of Fuchsian systems is one of Hilbert's 23 problems. In this paper we extend a basic result in scalar RH theory, the Sokhotski-Plemelj formula. Classically, this formula is derived under assumptions of H\"older continuity, although it also holds true under weaker, Dini, continuity conditions. Dini continuity is still too restrictive. We prove the Sokhotski-Plemelj formula under weaker assumptions, namely continuity at a point and an $L^1$ condition. Furthermore, we provide sufficient conditions for the existence of the Cauchy $\rm{P.V.}$ integral which are in a precise sense also necessary: weakening them runs into obstructions of a mathematical foundations nature.