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According to this site, the range is a subset of the codomain and represents what actually can be the output of a function. Why don't we have a name for the subset of the domain that can actually be the input of a function. Or, why my question makes no sense?

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    $\begingroup$ A function MUST be defined on the entire domain (otherwise it is a lie/"partial" function) $\endgroup$ – Alec Teal Sep 10 '15 at 21:21
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All of the domain is the input of the function. $f: \mathbb{N} \to \mathbb{R}$ means precisely that for every natural number $n$ there is a real $r_n$ which has $f(n) = r_n$. It makes no sense to say that $f$ is from $\mathbb{R} \to \mathbb{R}$, even though $\mathbb{N} \subset \mathbb{R}$, because to say that $f: \mathbb{R} \to \mathbb{R}$ is to say that for each real, $f$ takes a value on that real. In this example, there's just no way to say that $f(1/2)$ even has a value.

You may be looking for the idea of a partial function, which is not necessarily defined on its whole domain. We then say that the "domain of definition" of the partial function is the set consisting of everything on which the function really does take a value. For instance, the partial function $f: \mathbb{R} \to \mathbb{R}$ corresponding to the bona fide function $f: \mathbb{N} \to \mathbb{R}$ would have domain of definition $\mathbb{N}$, and domain $\mathbb{R}$.

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  • $\begingroup$ Yes, I was thinking of the case of partial functions. Thanks. $\endgroup$ – Lay González Sep 10 '15 at 20:10

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