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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.

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    Does not matter. Thanks for your hint.2011-09-22

2 Answers 2

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Hint (for $f: \mathbb{R} \to \mathbb{R}_\ell$): The continuous image of a connected set is connected. What are the connected components of $\mathbb{R}_\ell$?

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    @LostInMath: Thanks $f$or your detailed explaination.2011-09-22
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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.