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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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No topologies characterize differentiability as continuity
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by Robert Geroch, Erwin Kronheimer and George McCarty PDF
Proc. Amer. Math. Soc. 28 (1971), 273-274 Request permission

Abstract:

Do there exist topologies $\mathcal {U}$ and $\mathcal {V}$ for the set $R$ of real numbers such that a function $f$ from $R$ to $R$ is smooth in some specified sense (e.g., differentiable, ${C^n}$, or ${C^\infty }$) with respect to the usual structure of the real line if and only if $f$ is continuous from $\mathcal {U}$ to $\mathcal {V}$? We show that the answer is no.
References
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 28 (1971), 273-274
  • MSC: Primary 57.20; Secondary 26.00
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0271969-5
  • MathSciNet review: 0271969