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Most Infinitely Differentiable Functions are Nowhere Analytic

Published online by Cambridge University Press:  20 November 2018

R. B. Darst
R. B. Darst*
Affiliation:
Colorado State University, Fort Collins, Colorado
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We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 4 , December 1973 , pp. 597 - 598
Copyright
Copyright © Canadian Mathematical Society 1973

References

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2. Rudin, W., Principles of mathematical analysis, 2nd ed., McGraw-Hill, New York, 1964.Google Scholar
3. May, L. E., On C∞ functions analytic on sets of small measure, Canad. Math. Bull. 12 (1969), 25–30.Google Scholar