# Rigorous Definition of “Function of”

When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a distribution. I realized that while I know the definition of a function as a relation and I have an intuitive notion of what "function of" means, it's unclear to me how you transform this into a rigorous definition of "function of". So what is the actual definition of "function of"?

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The modern approach is, as you say, to view a function as a relation. Thus $f\subseteq A\times B$ is a function if it satisfies that if $(a,b)\in f$ and $(a,b')\in f$ then $b=b'$. It is then common to write $f(a)=b$ instead of $(a,b)\in f$.

This is a way to formalize the notion of $f$ defining its output as a function of its input. If you like then, this is the actual definition of 'function of'.

It is helpful to keep in mind the long history of the development of the notion of function. During the early days of the calculus a function $f:\mathbb R \to \mathbb R$ was vaguely defined to mean something like: f is a process that transforms the input $x$ to some output $f(x)$ and moreover $f$ does so in a very smooth way (almost always differentiable).

This historical approach to function, while not rigorous, is more in-line with $y$ being a function of $x$. The modern approach of a function as a relation, while very rigorous, is more static. This may be viewed as a shortcoming of this rigorous definition. However, the formalization of function is simple enough and easily allows abuse of concepts to actually think of a function as some process while it is formally not.

This situation is somewhat similar to the definition of a random variable. A random variable is nothing but a function with a particular domain and codomain. Thus, according to the relational definition, it is a very static thing. Nonetheless, we think of a random variable as a highly variable thing, even as if it's value is not yet known or is uncertain. However, this formalization of random variable within the rigorous confines of measure theory is highly useful, allowing one to correctly argue about uncertain events. This goes to show just how powerful the modern axiomatization is - there is enough flexibility in the interpretation of the notion of function to accomodate many situations.

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I'm not sure this answers my question. I still don't understand how from this we can determine whether a function is or is not a "function of" a given quantity. – MGriisser Dec 24 '12 at 21:52
formally, a function f is a function of x\in A precisely when f is a relation from A to some set B such that the axiom for function-hood is satisfied. – Ittay Weiss Dec 24 '12 at 21:54
thanks DonAntonio - I corrected it. – Ittay Weiss Dec 24 '12 at 22:01

"$y$ is a function of $x$" means the value of $y$ is determined by that of $x$. For example, to say that the area of a circle is a function of the radius implies that all circles with the same radius have the same area.

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+1. I'd add: it also implies that circles with different radii will generally have different areas. – ruakh Dec 25 '12 at 5:29
@ruakh : NO!!! It does not imply that at all! To say that is to say that the function is one-to-one. To say only that it's a function does not imply that it's one-to-one. – Michael Hardy Dec 25 '12 at 19:22
Michael's definition is definitely what it means in this context. But, a statistic in general, does not require the constraint given by @ruakh . This a restricted type of statistic (and function), in fact it is usually the case that statistics are given by surjective functions. – Lucas Dec 25 '12 at 19:26
@Lucas : Your point escapes me. What does surjectivity have to do with it? Whether a function is surjective depends on what the target set is. I don't see what it has to do with the present topic. – Michael Hardy Dec 25 '12 at 19:35
@MichaelHardy: That's why I said "will generally have". (My point being, you'd never say that the area of a unit circle is a function of the dimension of the n-space it's embedded in.) – ruakh Dec 25 '12 at 19:39

There certainly is a discrepancy between the formal set-theoretic definition ("giving" a function by giving its graph), and the informal use. Another important aspect of the informal use of "function" in practice is to ascertain when one thing $y$ is not "a function of" another thing $x$, which ordinarily means that "when $x$ changes", but everything else is "kept constant", $y$ does not change. A synonymous phrase is "$y$ does not depend on $x$".

How to ascertain whether $y$ "depends on/is a function of" $x$? There is no universal algorithm, and unless the relationship or lack thereof is described adequately, even specific examples are not resolvable. This is especially true of physical measurements, where correlation and causality are not always easy to distinguish.

In purely mathematical situations, often there is some difficulty in "finding" a thing $y$, and one is interested in being able to use "the same $y$" while other things in the environment/context vary. Giving upper bounds or lower bounds or counting something... with an outcome independent of, that is, not a function of, some other thing $x$... is a simpler story. It is not always obvious whether or not this is possible, so it is reasonable to ask the question.

In introductory physical science and engineering discussions, it is typically mathematically useful insofar as it simplifies things to assume (tentatively? heuristically? as a good approximation?) that one thing is independent of another, that is, "is not a function of". The archetype for this is a situation in which one will differentiate implicitly, but, if everything depends on all parameters, a uselessly complicated expression comes out. Using some experimental/physical sense about the physical realities often allows a practically useful approximation by declaring that this doesn't depend on that.

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A function $f$ is called "a function of $x$", if, for each $x$ (in some domain $X$), there is a unique corresponding output, denoted by $f(x)$.

So a statistic is a function of a sample from a distribution means that, given a sample $S$, a statistic takes that sample $S$ and spits out a unique statistic value $f(S)$.

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I would say "$f(x)$ is a function of $x$" or "$f$ is a function". – Michael Hardy Dec 25 '12 at 0:29

To answer this question, we must first ask ourselves "what is a variable?" What do I mean when I say that "$x$ is a real number-valued variable"?

I'm going to try and describe one useful approach.

We might think of $x$ as being a placeholder for an unknown but specific number. Or maybe a notation for expressing functions. But it is also useful to be able to consider the variable $x$ as simply being a real number, and not really any different from other real numbers like 0, 1, or $\pi$.

"But what is it's value?" you might ask. That's easy: it's value is $x$. "Is it positive, zero, or negative?" That one's easy too: the answer is "yes". Or more informatively, the truth value of the statement "$x$ is positive" is a variable too.

To distinguish modes of thought, let's reserve the term "real number" for the way we normally think, and use the term "scalar" to refer to real numbers in this new mode of thought.

If you can't wrap your head around this mode of thought, there are alternative semantics for this idea*: you can imagine there is some secret collection of "states", and every real number in this generalized sense is actually a real-valued function whose domain is the collection of states. e.g. in a physics context, the states might be the points in configuration space, and the scalars things like "temperature" or "the $x$-coordinate of the 17th particle".

The measure-theoretic notion of a random variable, or the analytic notion of a scalar field are very much examples of this sort of thing. (Which is why I chose the term "scalar")

Once you can wrap your head around scalars, you can imagine relationships between them. Just as $1$ and $2$ satisfy the relationship $1 + 1 = 2$, our real numbers $x$ and $y$ might satisfy the relationship $x + x = y$, or some more general sort of relationship $f(x,y) = 0$ for an ordinary function $f$. In this case, we say that $x$ and $y$ are functionally related. In the special case we can write $y = f(x)$, then we can say $y$ is a function of $x$.

(Why did I emphasize "ordinary" function? Just like it is useful to form the idea of $x$ being a variable number in the way I've described above, it is also useful to think of variable function in the same way; I wanted to emphasize that we are not doing that in the above paragraph)

If you are stuck thinking of scalars as functions of states, the notation $f(x,y)$ really means the function that sends the state $P$ to the number $f(x(P), y(P))$. A similar sort of composition happens when our scalars are random variables.

*: For those who know such things, I'm describing the internal logic of the topos of sheaves on a discrete space.

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I noticed lots of long answers so I'll give a concise answer here (this is not meant to be an insult to the other answers, or anything like that).

Let $A$ and $B$ be sets. A relation between $A$ and $B$ is some set $S \subseteq A \times B$. A function on $A$ is a relation between $A$ and $B$ where $B$ is an arbitrary set (call this relation $S \subseteq A \times B$), and if $(a,b) \in S$ and $(a,c) \in S$, then $b=c$.

For example, if we say $f$ is a function of time, and we take time to be any non-negative real number, then we have that $f$ is a subset of $\mathbb{R}_{\geq 0} \times A$ where $A$ is some arbitrary set.

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