Definition:Darboux Function

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Definition

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 $b$.

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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.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.4$: Continuity: Definition $1.4.1$