I wanted to know if it was true that if we are given a one-dimensional Feller process taking values in $\mathbb{R}$ and a hitting time $\tau_A=\inf\{t>0 s.t. X_t\in A\}$ with $A$ a open set (this to avoid measurability complexities but I would be also interested in the general case)
then if $X_0\in \bar{A}^c$ we have :
$\lim_{t\to 0}P[\tau_A
Regards