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 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 ). ${\displaystyle f\colon \mathbb {Z} /2\mathbb {Z} \to \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle f(x)=x^{2}-x}$ ${\displaystyle 0}$ ${\displaystyle 0^{2}-0=0}$ ${\displaystyle 1^{2}-1=0}$

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:

• 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 .