# Constant function

A constant real function of a variable ${\ displaystyle x}$

In mathematics , a constant function (from the Latin constans "fixed") is a function that always takes the same function value for all arguments .

## Definition and characterization

Let be a function between two sets . Then constant when all the following applies: . ${\ displaystyle f \ colon A \ to B}$${\ displaystyle f}$ ${\ displaystyle x, y \ in A}$${\ displaystyle f (x) = f (y)}$

The statement that the image set of consists of at most one element is equivalent to this definition . ${\ displaystyle f}$

In particular in category theory , constant functions are characterized by executing them one after the other:

${\ displaystyle f \ colon A \ to B}$is exactly constant if for all functions applies: .${\ displaystyle g, h \ colon C \ to A}$${\ displaystyle f \ circ g = f \ circ h}$

In this way constant morphisms are clearly defined. It is also common: If the link is constant for every function , then it is also constant. ${\ displaystyle g \ colon C \ to A}$${\ displaystyle f \ circ g}$${\ displaystyle f}$

## Properties, known functions

In the case of a constant function from the real numbers to the real numbers, its graph is a straight line that is parallel to the x-axis (“horizontal”) .

• If the function value is one , one often speaks of the one function . It is the derivation of identity .
The term “one function” is used in another context. A group structure can be defined on a number of functions by executing them one after the other. The neutral element of this group is often referred to as “one function”, but it is not a constant function, but the identical mapping .

The constancy of a function is not always obvious: If you look at any given function, it can be constant, although its function term apparently depends on the argument. One example is the function , i.e. on the remainder class ring modulo  2, using . This function is constant (da and ). ${\ displaystyle f \ colon \ mathbb {Z} / 2 \ mathbb {Z} \ to \ mathbb {Z} / 2 \ mathbb {Z}}$ ${\ displaystyle f (x) = x ^ {2} -x}$${\ displaystyle 0}$${\ displaystyle 0 ^ {2} -0 = 0}$${\ displaystyle 1 ^ {2} -1 = 0}$

## Further connections, generalizations

• The Liouville's theorem states that a limited , entire function is constant. It also follows from this that an elliptic function without a pole is constant.
• A generalization of constant functions are locally constant functions , where for each argument there is an environment around on which they are constant. This can be used to formulate the following sentences, for example: ${\ displaystyle x}$${\ displaystyle x}$
• Be a set that contains more than one element. A topological space is connected if every locally constant function is constant.${\ displaystyle Y}$ ${\ displaystyle X}$${\ displaystyle f \ colon X \ to Y}$
• Let be a continuous function between two topological spaces. If coherent and discrete is constant.${\ displaystyle g \ colon A \ to B}$${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle g}$

## literature

To the set-theoretical function concept:

• Paul Richard Halmos : Naive set theory . In: H. Kirsch, HG Steiner (ed.): Modern mathematics in elementary representation . 5th edition. Vandenhoeck & Ruprecht, Göttingen 1994, ISBN 3-525-40527-8 , pp. 43–47 (American English: Naive Set Theory . Translated by Manfred Armbust and Fritz Ostermann).

Constant functions in real and complex analysis:

In function theory, on Liouville's theorem:

• Heinrich Behnke , Friedrich Sommer: Theory of the analytical functions of a complex variable . Study edition, 3rd edition. Springer, Berlin / Heidelberg / New York 1972, ISBN 3-540-07768-5 .