For a filtration $\{\mathcal{F}_t\}_{t\in \mathbb{T}}$, the random variable $T$ is a stopping time if $\mathbb{1}[T\leq t]\in \mathcal{F}_t$ for all $t$. The textbook I'm reading emphasizes that in the discrete case $\mathbb{T}=\mathbb{N}$, this is equivalent to $\mathbb{1}[T=n]\in \mathcal{F}_n$ for all $n$. I believe that in the continuous case it is still true that $\mathbb{1}[T=t]\in\mathcal{F}_t$ for all $t$; it is just the converse that does not hold.
Indeed, for each $t$ the constant time $t$ is trivially a stopping time, and we have the general result that $\mathbb{1}[T=S]\in \mathcal{F}_{T\wedge S}$ for $T,S$ stopping times. Taking $S=t$ gives $\mathbb{1}[T=t]\in\mathcal{F}_{T\wedge t}\subseteq\mathcal{F}_{t}$ for all $t$. Is anything wrong with this argument?