Right continuous functions when considered with lower limit topology

Question

Exercise 17.7 (a) from James Munker's Topology says:

Suppose that $f : \mathbb R \rightarrow \mathbb R$ is "continuous from the right", i.e.
$$
\forall_{a \in \mathbb R} \lim_{x \rightarrow a^+} f(x) = f(a)
$$
Show that $f$ is continuous when considered as a function from $\mathbb R _{\mathcal l}$ to $\mathbb R$, where the former is the lower limit topology.

I have a cognitive dissonance. $f(x)=x$ is clearly continuous from the right. Take any open interval $(a, b)$. It's preimage is exactly $(a, b)$. Open sets in $\mathbb R _{\mathcal l}$ are unions of intervals of the form $[c, d)$. Either I can't see how $(a, b)$ can be a union of left-closed intervals, or my error it's elsewhere.

Answer 1

$$(a,b)=\bigcup_{c\in(a,b)}[c,b)$$