Editing Support (mathematics) (section)

==Compact support==

Functions with '''{{em|{{visible anchor|compact support}}}}''' on a topological space <math>X</math> are those whose closed support is a [[Compact space|compact]] subset of <math>X.</math> If <math>X</math> is the real line, or <math>n</math>-dimensional Euclidean space, then a function has compact support if and only if it has '''{{em|{{visible anchor|bounded support}}}}''', since a subset of <math>\R^n</math> is compact if and only if it is closed and bounded.

For example, the function <math>f : \R \to \R</math> defined above is a continuous function with compact support <math>[-1, 1].</math> If <math>f : \R^n \to \R</math> is a smooth function then because <math>f</math> is identically <math>0</math> on the open subset <math>\R^n \setminus \operatorname{supp}(f),</math> all of <math>f</math>'s partial derivatives of all orders are also identically <math>0</math> on <math>\R^n \setminus \operatorname{supp}(f).</math> 

The condition of compact support is stronger than the condition of [[Vanish at infinity|vanishing at infinity]]. For example, the function <math>f : \R \to \R</math> defined by
<math display="block">f(x) = \frac{1}{1+x^2}</math>
vanishes at infinity, since <math>f(x) \to 0</math> as <math>|x| \to \infty,</math> but its support <math>\R</math> is not compact.

Real-valued compactly supported [[smooth function]]s on a [[Euclidean space]] are called [[bump function]]s. [[Mollifier]]s are an important special case of bump functions as they can be used in [[Distribution (mathematics)|distribution theory]] to create [[sequence]]s of smooth functions approximating nonsmooth (generalized) functions, via [[convolution]].

In [[Well-behaved|good cases]], functions with compact support are [[Dense set|dense]] in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of [[Limit (mathematics)|limits]], for any <math>\varepsilon > 0,</math> any function <math>f</math> on the real line <math>\R</math> that vanishes at infinity can be approximated by choosing an appropriate compact subset <math>C</math> of <math>\R</math> such that
<math display="block">\left|f(x) - I_C(x) f(x)\right| < \varepsilon</math>
for all <math>x \in X,</math> where <math>I_C</math> is the [[indicator function]] of <math>C.</math> Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.