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I know that it is defined such that there is no more than one y-value output for any given x-value input, but I"m wondering WHY it is defined that way? Why can't we apply everything we know about functions to equations that have more than one y-value output for any given x-value input?

In short, what is the benefit or reason to define a function this way? At the same time, a little historical context would also be appreciated (i.e., when the concept originated and why).

Thanks,

Moshe

Amzoti
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Moshe
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  • http://en.wikipedia.org/wiki/Multivalued_function – Eckhard Jan 06 '13 at 16:28
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    Note that the "one value" constraint is less restrictive than you might think. For example a function $f: \mathbb{R} \to \wp(\mathbb{R})$ assigns a set of real numbers to any input. – WimC Jan 06 '13 at 16:30
  • @WimC Ermm ...but a function which maps a real to a set assigns a single value (a set) to a given argument. – Peter Smith Jan 06 '13 at 16:37
  • @PeterSmith, I think WimC is pointing out if the codomain can be varied - and specifically to the powerset of the OP's "y-values"- then any number of y-values can be assigned while still having a function. – alancalvitti Jan 06 '13 at 16:41
  • Of course, a function from $X$ to $\mathcal P^Y$ is the same thing as a relation between $X$ and $Y$. These are only the same thing as a multivalued function if you allow multivalued to include zero values, which is not usual. –  Jan 06 '13 at 16:56
  • The historical context is probably somewhere in the middle of Wikipedia's enormous article on the history of the function concept. –  Jan 06 '13 at 17:19
  • See also: http://en.wikipedia.org/wiki/Binary_relation, in the section "Special types of binary relations": a function is a (binary) relation that is functional and left-total. – alancalvitti Jan 06 '13 at 22:34
  • It has been a long time since I read Gerard Debreu's Theory of Value. If I recall correctly he used multi-valued function to prove the existence of a Pareto optimal solution. His proof used Kakutani's Fixed Point Theorem – Jay Jan 06 '13 at 22:45
  • One issue is that the abstract definition of a function as a left-total single-valued relation does not always match the use of the word "function" in the wild. And elementary calculus, in particular, tends to use traditional terminology. Recall that G.H. Hardy explained in his "Course in Pure Mathematics" that being defined for every input and giving only one value for every input were "by no means involved in the general idea of a function" (his words). This kind of traditional view of a function is still alive in elementary calculus and in physics and other applied areas. – Carl Mummert Sep 20 '17 at 16:41

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Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input. A function would not be well-defined if say $y = f(x)$ such that $y$ can take on any number of values at any particular input. Suppose the input $\;x = a\;$ outputs more than one distinct value, so that $y = f(a) \in\{y_1, y_2, ..., y_k, ...\} $. We'd never be able to say what, precisely, $y$ is when $x = a$.

If $y = f(a) = b$, and $y = f(a) = c$, and $\,b\ne c$, then the value of the function $\,\,f(x) = y\,$ at $\,a\,$ is ambiguous: $y$ would not specify any particular value at $a$. That is, its value is not well-defined at $a$, and perhaps not well-defined at other inputs, as well. Nor can we say much about the behavior of a function at a particular value, if it can take on many values at a given point.

E.g. How would we define continuity of, say, a real-valued multifunction?

The strict definition of a function, in terms of "outputing" exactly one value for any given input is really no more than an attempt to keeping functions well-defined, and thus properties of functions well-defined.

But you'd might like to explore the following:

See this entry on multi-valued "functions":

A multivalued function (shortly: multifunction, other names: many-valued function, set-valued function, set-valued map, multi-valued map, multimap, correspondence, carrier) is a left-total relation; that is, every input is associated with at least one output.

The term "multivalued function" is, therefore, a misnomer because functions are single-valued. Multivalued functions often arise from functions which are not injective. Such functions do not have an inverse function, but they do have an inverse relation. The multivalued function corresponds to this inverse relation. [bold-face mine]

Community
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amWhy
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  • Yet some texts in Complex Analysis do very cheerfully talk of many-valued functions ... E.g. Schaum's Outline of Complex Variables, 2ed (Schaum's Outline Series) by Murray Spiegel, Seymour Lipschutz, John Schiller, Dennis Spellman. This (2009 2nd edn. but first goes back to 1964: surely aims to be "middle of the road") explicitly talks about multi-valued functions (see §2.2, §2.6). – Peter Smith Jan 06 '13 at 16:30
  • Yes, indeed. I was referring to the "strict" definition students are usually introduced to. – amWhy Jan 06 '13 at 16:34
  • I wonder how the floor function is called function since for any value of x where x<x+1 all the results will be x. Any possible reason? – arqam Apr 18 '18 at 11:40
  • A function associates one, and only one, output to any particular input. I have often wondered why do we add the qualifier and only one to the definition of a function. It seems a little redundant. If we went ahead and left it out, what would happen. Specifically what is wrong with this definition: A function associates one output to any particular input. What conclusion would we get if we used this definition instead. – john Mar 16 '24 at 02:20
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Multi-valued functions do exist, but then that greatly limits our ability to say much about them. For functions, we're not just concerned about the value that it takes, but also the behavior of the function around the point. If we have multi-valued functions, then what we can say about the behavior of the point will be highly restricted, as it needs to be the same regardless of what value we choose.

A common example would be to let $f(x,y) = x^2 + y^2$, and ask for the values such that $f(x, y) = 1$ (i.e. the unit circle). The implicit function theorem is used, for us to conclude that there are 2 functions, namely $F(x) = \sqrt{1-x^2}$ and $F(x) = - \sqrt{1-x^2}$. If we had said that $F(x) = \pm \sqrt{1-x^2}$, it would be hard to talk about the behavior of $F'(x)$ exactly.

john
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Calvin Lin
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  • You may wish to read about Kakutani's Fixed Point Theorem. As I mentioned in another comment this result is used in Gerard Debreu's Theory of Value. This is a short book in mathematical economics. Anyone who knows undergraduate analysis should be able to read it. I will make no claim that the theory in this book applies to the real world. – Jay Jan 06 '13 at 23:02
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There is an interesting relevant post by Tim Gowers on his blog http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/

It is worth noting though, as a footnote, that the insistence that functions (properly so-called) be single-valued is a relative late-comer. G.H. Hardy, for example, in his once-canonical A Course in Pure Mathematics says that the characteristic of taking just one value for a given argument "is by no means involved in the general idea of a function".

Peter Smith
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  • Could you please shed some light on what that "general idea of a function" might be? I'm very curious, since the definition that Hardy is dismissing is the definition I eat and breathe when doing mathematics. – Ovi Dec 25 '18 at 05:51
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Obviously this limitation of defining a function f(x) as giving only ONE value of y is specific to the English (or American?) Maths. In French, we consider y^2 = x as a function, we call it horizontal parabole... See http://www.alloprof.qc.ca/BV/pages/m1330.aspx I would be tempted to use pejorative language toward English (or AMerican?) Maths!

Michel Macé
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  • But that's not true: "En mathématiques, une fonction est une relation entre un ensemble d’entrées (variable) et un ensemble de sorties (image), avec la propriété que chaque entrée est liée à exactement une sortie." – Trevor Gunn Jun 19 '17 at 03:54
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    The word "fonction" does not appear at all on the website you link to. What you have there are called "relations". – Trevor Gunn Jun 19 '17 at 03:58
  • Lol, postmodernist maths – AER Jun 29 '21 at 09:11