generalized Darboux function

Recall that a function $f:I\to\mathbb{R}$ , where $I\subseteq\mathbb{R}$ is an interval, is called a Darboux function if it satisfies the intermediate value theorem. This means, that if $a,b\in I$ and $f(a)\leqslant d\leqslant f(b)$ for some $d\in\mathbb{R}$ , then there exists $c\in I$ such that $a\leqslant c\leqslant b$ and $f(c)=d$ .

Darboux proved (see parent object) that if $f:[a,b]\to\mathbb{R}$ is differentiable then $f^{\prime}$ is a Darboux function. The class of Darboux functions is very wide. It can be shown that any function $f:\mathbb{R}\to\mathbb{R}$ can be written as a sum of two Darboux functions. We wish to give more general definiton of Darboux function.

Definition. Let $X$ , $Y$ be topological spaces. Function $f:X\to Y$ is called a (generalized) Darboux function if and only if whenever $C\subseteq X$ is a connected subset, then so is $f(C)\subseteq Y$ .

It can be easily proved that connected subsets of intervals (in $\mathbb{R}$ ) are exactly intervals. Thus this definition coincides with classical definiton, when $X$ is an interval and $Y=\mathbb{R}$ .

Note that every continuous map is a Darboux function.

Also the composition of Darboux functions is again a Darboux function and thus the class of all topological spaces, together with Darboux functions forms a category. The category of topological spaces and continuous maps is its subcategory.

Title	generalized Darboux function
Canonical name	GeneralizedDarbouxFunction
Date of creation	2013-03-22 19:18:36
Last modified on	2013-03-22 19:18:36
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	5
Author	joking (16130)
Entry type	Definition
Classification	msc 26A06