Blumberg Theorem

Motivation

It is immediately obvious that not every function f\colon \mathbb{R} \to \mathbb{R}, is continuous. However we can usually restrict the domain to remove any discontinuies. Even a function such as the Dirchlet function \chi_\mathbb{Q} can be restricted f \restriction \mathbb{Q} to create a continuous function.

Henry Blumberg proved in 1922 that for every function f \colon \mathbb{R} \to \mathbb{R} there exists a domain D such that f can be restricted to a continuous function.

This set D must be ‘big’ in some way to be interesting. Every function f\colon \mathbb{R} \to \mathbb{R} is continuous when restricted to \varnothing, or to some finite subset of \mathbb{R}.

Theorem

Theorem (Blumbergs Theorem)

For every f \colon \mathbb{R} \to \mathbb{R} there exists a dense subset D of \mathbb{R} such that f \restriction D is continuous.

Pleasant functions

Definition

A function f \colon \mathbb{R} \to \mathbb{R} is said to be f-pleasant if for every open interval I such that f(x) \in I, there exists an open interval J^I_x such that x \in J^I_x such that the set f^{-1}(B) is dense in J^I_x.