# 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 $f(a)\u2a7dd\u2a7df(b)$ for some $d\in \mathbb{R}$, then there exists $c\in I$ such that $a\u2a7dc\u2a7db$ 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 |