The complete question is from Mukres's Topology.
(a) Suppose that $f: \mathbb{R} \to \mathbb{R}$ is "continuous from the right" that is $\lim_{x \to a^{+}} f(x) = f(a),$ for each $a \in \mathbb{R}$. Show that $f$ is continuous when considered as a function from $\mathbb{R_\mathcal {l}}$ to $ \mathbb{R}$.
(b) Can you conjecture what functions $f: \mathbb{R} \to \mathbb{R}$ are continuous when considered as maps from $\mathbb{R}$ to $\mathbb{R_\mathcal {l}}$? As maps from $\mathbb{R_\mathcal {l}}$ to $\mathbb{R_\mathcal {l}}$?
NOTE:$\mathbb{R_\mathcal {l}}$ is the topology generated by the basis $\{[a,b)|a,b\in R\}$.
It is easy to prove the first part of the question. But I have no idea about how to figure out the second part of the question. Could you help me?
Thanks.