Suppose $X$ is 1-d stochastically continuous process with $X(0) = 0$, i.e.
$X\_s \to X\_t$ in probability as $s\to t$ for all $t\ge 0$. Let $\tau = \inf\{t>0: |X\_t|>1\}$. 


[Q.] Is $\tau>0$ almost surely?


I think the answer shall be yes, because $X$ has a Cadlag version (modification).
Thanks.



Hitting time of a stochastically continuous process
This is not true. Consider some random sequence that converges to $0$ a.s., for example, $Y\_n=(Z\_1+\cdots+ Z\_n)^{-1}$, where $Z$'s are i.i.d. (for instance) Exp(1) random variables. Set 
$$
X\_t = \begin{cases}
 2, & \text{if $t=Y\_k$ for some $k$},\\
 0, &\text{otherwise}.
 \end{cases}
$$
Then $X$ is stochastically continuous (since $X\_t=0$ a.s. for any fixed $t$), but $\tau=0$ a.s.


