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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

No topologies characterize differentiability as continuity


Authors: Robert Geroch, Erwin Kronheimer and George McCarty
Journal: Proc. Amer. Math. Soc. 28 (1971), 273-274
MSC: Primary 57.20; Secondary 26.00
MathSciNet review: 0271969
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0271969-5
PII: S 0002-9939(1971)0271969-5
Keywords: Differentiability
Article copyright: © Copyright 1971 American Mathematical Society