# Definition:Darboux Function

(Redirected from Definition:Intermediate Value Property)

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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 $f \left({a}\right)$ 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.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.4$: Continuity: Definition $1.4.1$