# Characterizing continuous functions from $N$ with discrete metric into $R$ with absolute value metric

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
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 $|f(x)-f(y)|=0<\epsilon$.
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