TITLE: Can you conjecture what functions $f: \mathbb{R} \to \mathbb{R}$ is continuous when considered as maps from $\mathbb{R}$ to $\mathbb{R_\mathcal{l}}$?
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: To get you started on $f:\mathbb{R}_\ell \to \mathbb{R}_\ell$: 
If $f$ is continuous as a function from $\mathbb{R}_\ell$ to $\mathbb{R}_\ell$, then it must be continuous as a function from $\mathbb{R}_\ell$ to $\mathbb{R}$, since every open set in $\mathbb{R}$ is also open in $\mathbb{R}_\ell$. Thus, as a function from $\mathbb{R}$ to $\mathbb{R}$ it must be continuous from the right. However, this isn’t enough: $f(x)=-x$ is continuous as a function from $\mathbb{R}$ to $\mathbb{R}$, but it’s not continuous as a function from $\mathbb{R}_\ell$ to $\mathbb{R}_\ell$. Why? If you can see what keeps this function from being continuous from $\mathbb{R}_\ell$ to $\mathbb{R}_\ell$, you’ve a good chance of working out exactly which functions from $\mathbb{R}_\ell$ to $\mathbb{R}_\ell$ are continuous.