Non-analytic smooth function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <a href="/facts/Smooth_function/mffL4ch2">smooth functions</a> (also called infinitely <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> functions) and <a href="/facts/Analytic_function/JhL3z8FP">analytic functions</a> are two very important types of <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>. One can easily prove that any analytic function of a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Argument_of_a_function/QkuPc28I">argument</a> is smooth. The <a href="/facts/Converse_(logic)/VhlUrxxx">converse</a> is not true, as demonstrated with the <a href="/facts/Counterexample/dnThYpHW">counterexample</a> below.
One of the most important applications of smooth functions with <a href="/facts/Compact_support/BTuMGbsb">compact support</a> is the construction of so-called <a href="/facts/Mollifier/aQx68Ksc">mollifiers</a>, which are important in theories of <a href="/facts/Generalized_function/fCM4VsyT">generalized functions</a>, such as <a href="/facts/Laurent_Schwartz/Kxelk3ju">Laurent Schwartz</a>'s theory of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a>.
The existence of smooth but non-analytic functions represents one of the main differences between <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> and <a href="/facts/Complex_manifold/DlvPnpzm">analytic geometry</a>. In terms of <a href="/facts/Sheaf_theory/MM3hcRw4">sheaf theory</a>, this difference can be stated as follows: the sheaf of differentiable functions on a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> is <a href="/facts/Fine_sheaf/ykaN5Ivq">fine</a>, in contrast with the analytic case.
The functions below are generally used to build up <a href="/facts/Partition_of_unity/4byX17aR">partitions of unity</a> on differentiable manifolds.

Non-analytic smooth function open-in-new

Non-analytic smooth function