Constant function: Difference between revisions

Revision as of 00:03, 14 January 2006

Revision as of 01:21, 14 January 2006

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The first characterization of constant functions given above, is taken as the motivating and defining property for the more general notion of [[constant morphism]] in [[Category theory]].

In contexts where it is defined, the [[derivative]] of a function measures how that function varies with respect to the variation of some argument. It follows that, since a constant function does not vary, its derivative, where defined, will be zero. Thus for example:

*If ''f'' is a [[real number|real-valued]] function of a real [[variable]], defined on some [[interval]], then ''f'' is constant if and only if the [[derivative]] of ''f'' is everywhere zero.

For functions between [[preorder|preordered sets]], constant functions are both [[order-preserving]] and [[order-reversing]]; conversely, if ''f'' is both order-preserving and order-reversing, and if the [[___domain]] of ''f'' is a [[lattice (order)|lattice]], then ''f'' must be constant.

Other properties of constant functions include:

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* Every constant function between [[topological space]]s is [[continuous function (topology)|continuous]].

For functions between [[preorder|preordered sets]], constant functions are both [[order-preserving]] and [[order-reversing]]; conversely, if ''f'' is both order-preserving and order-reversing, and if the [[___domain]] of ''f'' is a [[lattice (order)|lattice]], then ''f'' must be constant.

In a [[connected set]], the function is [[locally constant]] if and only if it is constant.

== Derivative ==

In contexts where it is defined, theThe [[derivative]] of a function measures how that function varies with respect to the variation of some argument. It follows that, since a constant function does not vary, its derivative, where defined, will be zero. ThusIndeed, suppose f(''x'') = ''c'' for example:all vector ''x'' in an Euclidean space and ''c'' constant. Then, for <math>h \ne 0</math>

:<math>\left| {f(x + h) - f(x) \over h} \right| = \left| {c - c \over h} \right| = 0</math>.

It turned out the converse follows when ''f'' is real-valued; since the [[mean-value theorem]] says <math>|f(b) - f(a)| = |f'(x)|</math> for ''f'' differentiable on [''a'', ''b''] and some ''x'' between. The converse also holds when ''f'' maps real vectors, provided that all of its partial derivatives vanish. If only some of the partial derivatives are zero, then ''f'' is [[locally constant]]. If, however, the function is [[locally constant]] on a connected set, then it is constant.

==References==