Turning Automatic Continuity Around: Automatic Homomorphisms

Abstract

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 $G=H=\mathbb{R}$.