When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$? Let $E$ be a locally compact Hausdorff space with countable base and $X_t$ be a stochastic process taking values in the one-point compactification of $E$ (with the Borel $\sigma$-algebra).  Let $f$ be a continuous function vanishing at infinity.  I'm wondering under what conditions it is true that $x \mapsto \mathbf{E}^{x}[f(X_t)]$ is a continuous function?
If $X_t$ is a Brownian motion on $\mathbb{R}$, it is straightforward to verify this is true and, in fact, it is true whenever $X_t$ is an Ito diffusion.
An example where this fails is if we let $E=\mathbf{R}$ and let $X_t$ be a reflected Brownian motion on $\{ x: x\geq 0 \}$ and be the negative of the absolute value of a Brownian motion in $\mathbb{R}^3$ on $\{x : x<0 \}$.
More broadly, I'm curious as to what kind of conditions on the sample paths of a Markov process $X$ with continuous paths force it to be a Feller process.  The condition I'm asking about seems to be the one that does not come for free if you start with such an $X$.
 A: UPDATE: This is at best a partial answer.
If $X$ is a Markov process with continuous paths, then the additional condition for it to be a diffusion it be strong Markov (see the book by Ito-McKean, Section 3.1). For example, your example is not strong Markov (stop it when it hits 0). In that case, however, it is still not necessarily Feller (this was stated previously).
If you only require $X$ to be a Markov process with not necessarily continuous paths, then your condition is (almost, see this thread) equivalent to requiring that 1) $S_t:C_0(E) \rightarrow C_0(E)$, where $(S_t)_{t\ge0}$ is the semigroup of $X$ and $C_0(E)$ is the space of continuous functions vanishing at infinity. But a Feller process requires moreover that 2) $S_tf(x) \rightarrow f(x)$ for each $x$, as $t\rightarrow 0$ (see for example Section III.3 in the book by Revuz-Yor). For example, take Brownian motion on $\mathbb R^+$ which jumps to $1$ when it hits $0$. This process satisfies 1) but not 2)
If you take for $X$ any stochastic process, I doubt that one can tell more than what you wrote in your comment.
