Definition of a function of a real variable

Question

Wikipedia says that

a function of a real variable is a function whose domain is the real numbers $\mathbb R$, or a subset of $\mathbb R$ that contains an interval of positive length.

I don't understand what it means for a subset of $\mathbb R$ to have an interval of positive length. Does this mean that the subset cannot have only integers since this particular subset would be just points on a number line and not a length on the number line? What if the subset of $\mathbb R$ was $\{1,2\}\cup[3,4]$? Is this a valid subset? It has 2 points and a length on the number line but the only condition is that the subset has an interval of positive length so it seems that this is a valid subset of $\mathbb R$. Also, what is the point of saying "positive length"? Why can't the definition of a function of a real variable just say "length"?

Have you seen measure theory? Measure theory rigorously defines what it means for a set to have positive "length" or measure, and is only undefined on very pernicuous sets. — Rushabh Mehta, Oct 18 '19 at 01:07
The set you proposed has length $1$. So yes, it is valid. They simply don't want to deal with sets of length 0. — Rushabh Mehta, Oct 18 '19 at 01:07
The definition is stipulating positive length as opposed to zero length. The interval $[a,a] = {a}$ is an example of a (degenerate) interval of zero length. — , Oct 18 '19 at 01:12
Yes, that is valid. What wouldn't be valid is a domain of $\mathbb Q$. An interval "of positive length" means there is some $x,y; x< y$ so that $(x,y)$ is a subset of domain. Now you might wonder how there can be an interval of zero or negative length and that's fair. But some text would consider the set ${x}$ to be an interval of zero length (depends on the text). I think to idea is that the real numbers, as opposed to the radicals has continuum and so a real function must have some continuum interval; it can be a really small space of continuum but it must exist. — fleablood, Oct 18 '19 at 01:21

score 0 · Accepted Answer · answered Oct 18 '19 at 01:22

This is the definition provided on Wikipedia.

A function of a real variable is a function whose domain is the real numbers $\mathbb R$, or a subset $D\subseteq\mathbb R$, which contains an interval of positive length.

This means that the function $f$ is called a function of a real variable if and only if $f:D\to E$ for some set $E$ (common examples are the real numbers $\mathbb R$ or the complex numbers $\mathbb C$), where the domain $D$ is a subset of $\mathbb R$ and in addition, $D$ has a subset $(a,b)\subseteq D$ so that $b>a$. Here, the "length" of an interval of the forms $(a,b),[a,b],[a,b),(a,b]$ are all defined to be $b-a$. So by "positive length" we mean that the length $(b-a)>0$, i.e. $b>a$. The reason we need to make this definition is because usually, we consider sets of the form $[a,a]=\{ a\}$ as intervals too. But of course, this set has zero length.

In your example, the domain $D=\{1,2\}\cup[3,4]$ does satisfy this criterion, because the interval $[3,4]\subset D$ is an interval of positive length (in particcular, it has length $1>0$). On the other hand, sets like the rational numbers $\mathbb Q$ but also more "pathological" sets like the Cantor set can be shown to contain no intervals of positive length, and will not therefore suffice as an example of the domain of a real variable, as it is defined on Wikipedia. The formal definition of this is in terms of measures, specifically the Lebesgue measure, which formalises the idea of lengths. The Lebesgue measure of the Cantor set is $0$ [see here], so that it does not contain an interval of positive length.

The reason we often impose the restriction that $D$ has an interval of positive length is because we often want to study changes in $f$ around the neighbourhood of a particular point $x$ in the domain. We want to ask questions you might have seen in Calculus, such as "if we change $x$ by a little bit, how much does $f(x)$ change?". We can do so if there is a connected set (i.e. an interval) to deal with. But if our set looks like $D=\mathbb Q$ for example, it makes our life harder. So we exclude those sets from definition just so that we don't need to think about them.

Definition of a function of a real variable

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