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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.2012-12-10

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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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    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.2012-12-10