generalized Darboux function

Recall that a function f:I, where I is an interval, is called a Darboux function if it satisfies the intermediate value theorem. This means, that if a,bI and f(a)df(b) for some d, then there exists cI such that acb and f(c)=d.

Darboux proved (see parent object) that if f:[a,b] is differentiableMathworldPlanetmathPlanetmath then f is a Darboux function. The class of Darboux functions is very wide. It can be shown that any function f: 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 spacesMathworldPlanetmath. Function f:XY is called a (generalized) Darboux function if and only if whenever CX is a connected subset, then so is f(C)Y.

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

Note that every continuous mapMathworldPlanetmath 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