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