# Constant function

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: .

The statement that the image set of consists of at most one element is equivalent to this definition .

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

- is exactly constant if for all functions applies: .

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.

## 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 value of the function is the number zero , it is a special case of the zero function (or zero mapping). In both real and complex differential calculus , the derivative of a constant function is the null function. If you define a vector space structure on a set of functions, the null function always corresponds to the null vector .

- 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 .

- Polynomials of the zeroth degree are constant functions. A constant function between vector spaces is a linear mapping if and only if it is the null function.

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 ).

## 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:
- Be a set that contains more than one element. A topological space is connected if every locally constant function is constant.
- Let be a continuous function between two topological spaces. If coherent and discrete is constant.

## 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:

- Harro Heuser : Textbook of Analysis . 8th edition. Part 1. BG Teubner, Stuttgart 1988, ISBN 3-519-12231-6 .

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 .