Many books [see below for references] explore the connections between partial differential equations and expectation values.

Assume $X$ is a diffusion with generator $A$, then they conclude, that under certain conditions, the function $u(t,x):=E^{t,x}(f(X_T))$ is the solution to the following Cauchy problem:

$$\begin{gather} \frac{\partial u}{\partial t}+Au=0, \text{ on } [t,T)\times\mathbb R \\ u(T,x) = f(x), \text{ on } \mathbb R\end{gather} \tag{1}$$

More genereal versions of this are usually called the Feynman-Kac theorem.

**Questions**

Do you have a reference where $f$ is not required to be continuous?

I have never (expect for the case where $X$ is a Brownian Motion) found any theorem that did not require $f$ to be continuous. Why?

Apart from rigorous treatments (as in the mentioned books), most people seem not to care about $f$'s continuity. (e.g. the Wikipedia entry on the Kolmogorov backward equations, and many more). Are they wrong?

What about piecewise continous $f$? Is the problem to trivial and to specialized for the cited books?

**My understanding**

If for the process $X$ has a transition density $p(t,x,s,y)$ that is in $C^{1,2}$ for fixed $s,y$ (which seems to be the case under similar conditions as for the Feynman-Kac theorem), then

$$ u(t,x) = \int f(y) p(t,x,T,y) dy $$

and $u$ is also in $C^{1,2}$ by Leibniz's integral rule for bounded $f$ and thus should also solve (1).

Is this correct?

*References*

Oksendal, Stochastic Differential Equations

Karatzas and Shreve, Brownian Motion and Stochastic Calculus

Friedman, Stochastic differential equations and applications