On Everywhere Surjective Functions

Abstract: Analysis is full of examples verifying some kind of pathological property — continuous nowhere differentiable functions; discontinuous linear functions; differentiable nowhere monotone functions, etc. In this talk, we will focus on another interesting example — everywhere surjective functions. 

A function f from R to R is called everywhere surjective, if for every non-empty interval (a, b), the image f((a, b)) is the entire R. Such function is doomed to be discontinuous at every real number, and, in particular, provides a “strong” counterexample to the converse of the Intermediate Value Theorem. Namely, f satisfies the Intermediate Value Property — for any two numbers x1<x2 in R, if y is a real number between f(x1) and f(x2), then there exists a real number c in (x1, x2) such that f(c)=y — yet f is not even close to being continuous.

The existence of everywhere surjective functions was first noticed by H. Lebesgue. In fact, there are plenty of such functions. The set S of everywhere surjective functions on R is 2^c-lineable; that is, SU{0} contains a vector subspace of the largest possible dimension, 2^c.

 We will consider several examples, and look into the properties of these functions.