I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous, by means of solving a PDE or Feynman-Kac equation.
Edit adressing the comments: Even if the PDE has a solution it can only be shown to equal the expectation under certain conditions. That is why I am asking for a reference for the caclulation of the expectation as solution to a PDE and not about the PDE and its solutions in itself.
Any such "verification theorem" basically uses the Ito formula for the value function and thus requires twice differentiability. This can only be ensured for the PDE solotion if the end data is continuous. So I am not interested in "it should just work" arguments but rather in answers or references to the "when" and "why.