Limit probability for some Hitting time of a Feller Process 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 :
$\lim_{t\to 0}P[\tau_A<t]=0$
Regards

Some useful facts about Feller processes are at http://almostsure.wordpress.com/2010/07/19/properties-of-feller-processes/.
In particular, every Feller process admits a cadlag modification.  If you are willing to assume you are dealing with such a modification, then your question becomes pretty easy, since by continuity of probability $\lim_{t \to 0} P(\tau_A < t) = P(\tau_A = 0)$.  But $\bar{A}^c$ is open and $X_t$ is a.s. continuous at 0 so this has probability 0.  It doesn't much matter that $A$ be open or even measurable, since if you replace $A$ by $\bar{A}$ the answer is the same.
If you insist on using a non-cadlag modification, I am not sure that you can say much.
