I'm struggling to understand how the continuity of a standard Brownian motion is affected by the addition of the drift. Let me explain my doubts by means of the following question.
Consider a Brownian motion with constant drift $\mu$ and scale parameter $\sigma$: $$X_t = \mu t + \sigma Z_t$$ where $Z_t$ is a standard Brownian motion.
Let $r \in R$ be a scalar and consider the indicator function $1_{[r, \infty)}(X_t)$ that takes value one if $X_t \in [r, \infty)$ and zero otherwise.
What is the expectation $E\{1_{[r, \infty)}(X_{t+\Delta})|X_t\}$ of the indicator function at $X_{t+\Delta}$ conditional on $X_t$, for $\Delta > 0$?
What happens in the limit for $\Delta \rightarrow 0$?
My basic intuition would be that in the limit the expectation becomes degenerate, taking values one and zero depending on whether $X_t \in [r, \infty)$ or not, but it might well be that the drift changes things. Is the intuition correct?