Editing Smoothness (section)

===Smooth partitions of unity===
Smooth functions with given closed [[Support (mathematics)|support]] are used in the construction of '''smooth partitions of unity''' (see ''[[partition of unity]]'' and [[topology glossary]]); these are essential in the study of [[smooth manifold]]s, for example to show that [[Riemannian metric]]s can be defined globally starting from their local existence. A simple case is that of a ''[[bump function]]'' on the real line, that is, a smooth function ''f'' that takes the value 0 outside an interval [''a'',''b''] and such that
<math display="block">f(x) > 0 \quad \text{ for } \quad a < x < b.\,</math>

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals <math>(-\infty, c]</math> and <math>[d, +\infty)</math> to cover the whole line, such that the sum of the functions is always 1.

From what has just been said, partitions of unity do not apply to [[holomorphic function]]s; their different behavior relative to existence and [[analytic continuation]] is one of the roots of [[Sheaf (mathematics)|sheaf]] theory. In contrast, sheaves of smooth functions tend not to carry much topological information.