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 $\map f a$ and $\map f b$, there exists a $c \in S$ such that $a \le c \le b$ and $\map f c = y$.
That is, for every intermediate value between $\map f a$ and $\map f b$, 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.