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Non-analytic smooth function

In real analysis, a smooth function is infinitely differentiable at each point in its domain, while a real analytic function is, at each point in its domain, the limit of a convergent power series in a neighbourhood of that point. All real analytic functions are smooth, but there exist smooth real functions that are not real analytic, as given below.

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An example function

A smooth function that is nowhere real analytic

Application to Taylor series

Application to higher dimensions

Complex analysis

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Non-analytic smooth function

Nodes67
Edges66
Triples0
Avg. degree1.97
Density0.029851
Components1

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function analytic smooth functions real derivatives differentiable series origin every displaystyle mathbb hence infinitely point support complex one transition taylor

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