generalized Darboux function

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

Darboux proved (see parent object) that if $f:[a,b]\to ℝ$ is differentiable then $f^{\prime }$ is a Darboux function. The class of Darboux functions is very wide. It can be shown that any function $f:ℝ\to ℝ$ 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 $ℝ$) are exactly intervals. Thus this definition coincides with classical definiton, when $X$ is an interval and $Y=ℝ$.

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.