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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
DOI: https://doi.org/10.1090/S0002-9939-1971-0271969-5
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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Keywords: Differentiability
Article copyright: © Copyright 1971 American Mathematical Society