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How can we characterize all continuous functions from $N$ with the discrete metric into $R$ with the absolute value metric?

I'm not sure what the question is asking. Can anyone elaborate?

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Basically, what information can you gain about a function if all you know is that it is a continuous $f:\mathbb{N} \rightarrow \mathbb{R}$ with the given metrics. –  Tom Oldfield Dec 10 '12 at 0:15
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1 Answer 1

up vote 3 down vote accepted

When you are asked for a characterisation, you need to state conditions that are equivalent to what is given or at least imply them or are implied by them.

Any function from $\mathbb N$ with the discrete metric to $\mathbb R$ with the absolute value metric will be continuous! This is because at any point $x$ in the domain, for any $\epsilon>0$, $d(x,y)<1$ implies that $x=y$ so that $|f(x)-f(y)|=0<\epsilon$.

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Is it sufficient to show the case when $x=y$, or would I need to show when they are not equal as well? –  Alti Dec 10 '12 at 0:40
    
Ohh okay, I got a little confused with the quantifiers. I see what you mean now, thanks! –  Alti Dec 10 '12 at 0:47
    
Alternatively: a function is continuous iff the inverse image of every open subset of the target space is open in the domain. But this domain has the property that every subset is open. Thus every function is continuous. –  Lubin Dec 10 '12 at 0:56
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