Definition:Darboux Function

From ProofWiki
Jump to navigation Jump to search


Let $S \subseteq \R$.

Let $f: S \to \R$ be a real function.

Then $f$ is Darboux if and only if, given any $a, b \in S$ and $y \in \R$ such that $a < b$ and $y$ is between $f \left({a}\right)$ and $f \left({b}\right)$, there exists a $c \in S$ such that $a \le c \le b$ and $f \left({c}\right) = y$.

That is, for every intermediate value between $f \left({a}\right)$ and $f \left({b}\right)$, that value is the image of some intermediate value between $a$ and $b$.

Also known as

$f$ is said to have the intermediate value property (frequently abbreviated as I.V.P.) if and only if $f$ is a Darboux function.

Source of Name

This entry was named for Jean-Gaston Darboux.