About: Non-analytic smooth function

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In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The 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 mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.

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rdfs:label	Non-analytic smooth function (en)
rdfs:comment	In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The 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 mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions. (en)
sameAs	Non-analytic smooth function Non-analytic smooth function
dbp:wikiPageUsesTemplate	dbt:Clear dbt:Main dbt:Math dbt:Reflist dbt:Short_description dbt:Collapse_bottom dbt:Collapse_top dbt:Planetmath_reference
Subject	Articles containing proofs Smooth functions
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Non-analytic_smooth_function?oldid=1053659906&ns=0
Wikipage page ID	421463 (xsd:integer)
page length (characters) of wiki page	13829 (xsd:nonNegativeInteger)
Wikipage revision ID	1053659906 (xsd:integer)
Link from a Wikipage to another Wikipage	Fourier series Smoothness Support (mathematics) Uniform convergence Uniform norm Unit interval 0 Analytic continuation Analytic function Analyticity of holomorphic functions Ball (mathematics) Borel's lemma Monomial Radius of convergence Taylor series Bump function Articles containing proofs Smooth functions Cauchy–Hadamard theorem Generalized function Differentiable function Smoothness Differential geometry Distribution (mathematics) Picard theorem Chain rule Degree of a polynomial Sheaf (mathematics) Complex analysis Complex manifold Complex plane Continuous function Converse (logic) Counterexample Dyadic rational One-sided limit Origin (mathematics) Partition of unity Flat function Function (mathematics) Derivative Exponential function Extreme value theorem Fabius function Injective sheaf Integer Mollifier Weierstrass M-test Essential singularity Euclidean space Émile Borel Laurent Schwartz Mathematical induction Mathematics History of numbers Polynomial Ratio test Real line Real number Reciprocal rule Recursion Sign (mathematics)
has abstract	In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The 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 mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions. The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and 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, in contrast with the analytic case. The functions below are generally used to build up partitions of unity on differentiable manifolds. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Non-analytic_smooth_function
is Wikipage redirect of	Non-analitic smooth function An infinitely differentiable function that is not analytic An infinitely differentiable function that is not analytic/proof Infinitely-differentiable function that is not analytic
is Link from a Wikipage to another Wikipage of	Analytic function Non-analitic smooth function Bump function List of mathematical examples List of real analysis topics Earle Raymond Hedrick Flat function Mollifier Klein–Kramers equation An infinitely differentiable function that is not analytic

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