\begin{definition}[Definition:Darboux Function]
Let $S \subseteq \R$.
Let $f: S \to \R$ be a real function.
Then $f$ is '''Darboux''' {{iff}}, 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 $b$.
\end{definition}