The novice, through the standard elementary mathematics indoctrination, may fail to appreciate that, compared to the natural, integer, and rational numbers, there is nothing simple about defining the real numbers. The gap, both conceptual and technical, that one must cross when passing from the former to the latter is substantial and perhaps best witnessed by history. The existence of line segments whose length can not be measured by any rational number is well-known to have been discovered many centuries ago (though the precise details are unknown). The simple problem of rigorously introducing mathematical entities that do suffice to measure the length of any line segment proved very challenging. Even relatively modern attempts due to such prominent figures as Bolzano, Hamilton, and Weierstrass were only partially rigorous and it was only with the work of Cantor and Dedekind in the early part of the 1870’s that the reals finally came into existence.Jeremy Avigad writes:
Today, we think of a “function” as a correspondence between any two mathematical domains: we can talk about functions from the real numbers to the real numbers, functions that take integers to integers, functions that map each point of the Euclidean plane to an element of a certain group, and, indeed, functions between any two sets. As we began to explore topic, Morris and I learned that most of the historical literature on the function concept focuses on functions from the real or complex numbers to the real or complex numbers. ...It is funny because it so clumsy. It should be obvious that a function can have any domain, and have any definition on that domain.
Even the notion of a “number theoretic function,” like the factorial function or the Euler function, is nowhere to be found in the literature; authors from Euler to Gauss referred to such entities as “symbols,” “characters,” or “notations.” Morris and I tracked down what may well be the first use of the term “number theoretic function” in a paper by Eisenstein from 1850, which begins with a lengthy explanation as to why is it appropriate to call the Euler phi function a “function.” We struggled to parse the old-fashioned German, which translates roughly as follows:
Once, with the concept of a function, one moved away from the necessity of having an analytic construction and began to take its essence to be a tabular collection of values associated to the values of one or several variables, it became possible to take the concept to include functions which, due to conditions of an arithmetic nature, have a determinate sense only when the variables occurring in them have integral values, or only for certain value-combinations arising from the natural number series. For intermediate values, such functions remain indeterminate and arbitrary, or without any meaning.When the gist of the passage sank in, we laughed out loud.
It is easy to forget how subtle these concepts are, as their meanings have been settled for a century and explained in elementary textbooks. But they were not obvious to some pretty smart 19th century mathematicians. Even today, most physicists and philosophers have never seen rigorous definitions of these concepts.
Now a function can be defined as a suitable set of ordered pairs, once set theory machinery is defined. The domain of the function can be any set, and so can the range. These things seem obvious to mathematicians today, but it took a long time to get these concepts right. And concepts like infinitesimals are still widely misunderstood.