3,233 research outputs found
No functions continuous only at points in a countable dense set
We give a short proof that if a function is continuous on a countable dense
set, then it is continuous on an uncountable set. This is done for functions
defined on nonempty complete metric spaces without isolated points, and the
argument only uses that Cauchy sequences converge. We discuss how this theorem
is a direct consequence of the Baire category theorem, and also discuss
Volterra's theorem and the history of this problem. We give a simple example,
for each complete metric space without isolated points and each countable
subset, of a real-valued function that is discontinuous only on that subset.Comment: Expanded the result and added historical references and discussio
Measurable Dynamics of Maps on Profinite Groups
We study the measurable dynamics of transformations on profinite groups, in
particular of those which factor through sufficiently many of the projection
maps; these maps generalize the 1-Lipschitz maps on .Comment: 18 page
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