Non-analytic smooth function

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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. 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. Smooth real functions with domain R n {\displaystyle \mathbb {R} ^{n}} and with support of compact closure (bump functions) are non-analytic at each boundary point of the closure of its support. One of the...

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