Real Analysis Exchange

Turning Automatic Continuity Around: Automatic Homomorphisms

Ryan M. Berndt and Greg G. Oman

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Let $G$ and $H$ be Polish groups and let $\pi\colon G\rightarrow H$ be a function. The automatic continuity problem is the following: assuming $\pi$ is a group homomorphism, find conditions on $G$, $H$, or $\pi$ which imply that $\pi$ is continuous. In this note, we initiate a study of a reverse problem: supposing $\pi$ is continuous, find conditions on $G$, $H$, or $\pi$ which imply that $\pi$ is a homomorphism. Herein, we treat the case $G=H=\mathbb{R}$.

Article information

Real Anal. Exchange, Volume 41, Number 2 (2016), 271-286.

First available in Project Euclid: 30 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22A05: Structure of general topological groups 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
Secondary: 20F38: Other groups related to topology or analysis

automatic continuity homomorphism lattice Polish group


Berndt, Ryan M.; Oman, Greg G. Turning Automatic Continuity Around: Automatic Homomorphisms. Real Anal. Exchange 41 (2016), no. 2, 271--286.

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