Constant function

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

Definition and characterization

Let be a function between two quantities. Then is constant if and only if for all.

Equivalent to this definition is the statement that the image consists of a maximum quantity of an item.

In particular, in the category theory constant functions are characterized by sequential execution:

In this way, constant morphisms are defined clean. Commonly remains: Is for each function linking constant, is also constant.

Properties, known functions

In the case of a constant function from the reals to the reals its graph is a parallel to the x- axis ( " horizontal " ) line.

• Is the value of the function, the number zero, is the special case of zero function (or zero figure). Both the real and the complex differential calculus, the derivative of a constant function is the zero function. If we define a vector space structure on a set of functions, the zero function is always equal to the zero vector.

• If the function value one, so one often speaks of the one function. Is the derivative of the identity.

• Polynomials of degree zero are constant functions. Between vector spaces is a constant function is a linear map if and only if it is the zero function.

The constancy of a function is not always obvious: If we consider an arbitrarily given function, it may be constant, although their function term apparently depends on the argument. An example is the function that is on the residue class ring modulo 2 means. This function is constant ( and since ).

Other relationships, generalizations

• The set of Liouville states that a bounded entire function is constant. It also follows that an elliptic function without pole is constant.
• A generalization of constant functions are locally constant functions for which an environment to exist for each argument on which they are constant. This allows, for example, the following sentences formulated: Is a set containing 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. Is continuous and discrete, so is constant.