Editing Non-analytic smooth function (section)

{{Short description|Mathematical functions which are smooth but not analytic}}
In [[mathematics]], [[smooth function]]s (also called infinitely [[Differentiable function|differentiable]] functions) and [[analytic function]]s are two very important types of [[function (mathematics)|functions]]. One can easily prove that any analytic function of a [[real number|real]] [[Argument of a function|argument]] is smooth. The [[converse (logic)|converse]] is not true, as demonstrated with the [[counterexample]] below.

One of the most important applications of smooth functions with [[compact support]] is the construction of so-called [[mollifier]]s, which are important in theories of [[generalized function]]s, such as [[Laurent Schwartz]]'s theory of [[distribution (mathematics)|distribution]]s.

The existence of smooth but non-analytic functions represents one of the main differences between [[differential geometry]] and [[complex manifold|analytic geometry]]. In terms of [[sheaf theory]], this difference can be stated as follows: the sheaf of differentiable functions on a [[differentiable manifold]] is [[fine sheaf|fine]], in contrast with the analytic case.

The functions below are generally used to build up [[partition of unity|partitions of unity]] on differentiable manifolds.