Let $X=\{X_t\}_{t\geq 0}$ be a weak solution of $$ dX_t = \mu(t,X)\,dt+\sigma(t,X)\,dW_t $$ where $\mu$ and $\sigma$ are measurable and W is a Brownian motion. In this paper it is said that $$ \inf\{t>0;X_t>X_0\} = \inf\{t>0;X_t<X_0\} =\inf\{t>0;X_t=X_0\}=0,~~a.s. $$ To verify this, the reader is ask to look at the book by Karatzas and Shreve at 2.7.18. Here, however, I found only the statement of the following problem:
which does not help me much. I am therefore looking for a more didactic reference and any suggestion is welcome.