Continuity of Optimal Stopping Time as a Function of the Starting Point

Question

Consider the time-homogeneous optimal stopping problem $$V(x) = \sup_{\tau} E(f(X_\tau^x))$$ where $X^x$ is a diffusion starting at the point $x \in \mathbb{R}$. Typically, optimal stopping problems have an associated optimal stopping time $$\tau(x) \equiv \inf \{t \ge 0 : X^x_t \in S\}$$ where $S \equiv \{y: V(y) = f(y)\}$ under appropriate conditions on $f$ and $V$ (see, for example, Peskir and Shiryaev in Chapter 1). I would like to understand under what conditions (if any) is $\tau(x)$ continuous as a function of $x$. Any references/insight would be massively appreciated!