Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

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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.

Thank you