In a book I read the following sentence:
"Let $V$ be a countable set, equipped with the discrete topology. A function $f : [0, \infty) \rightarrow V$ is right-continuous if $\lim_{s \downarrow t} f(s) = f(t)$. Since $V$ is equipped with the discrete topology, we have that $f$ is right-continuous if and only if for every $t \geq 0$ there exists $\epsilon >0$ such that $f(s) = f(t)$ for every $s \in [t, t+\epsilon]$."
My question is: why does the if and only if condition stated in the second sentence follow from the fact that we have the discrete topology? What is then the definition of $\lim_{s \downarrow t} f(s) = f(t)$ in the discrete topology?
The singleton set $\{f(t)\}$ is an open set in the discrete topology. Hence the condition $f(s) \to f(t)$ as $s$ decreases to $t$ is equvalent to the condition $f(s)=f(t)$ whenever $t <s< t+\delta$ for some $\delta >0$.