---
tag: A7549DJW1N
kind: definition
name: Darboux function
msc: [26A15, 26A24]
defines: 7BJEH1GCTE
uses: [0TK8Y3J5RW, 0XHDTE5YMZ, 0P27W9RW0G, 00KAFBR32E, 0Q5A47J48D, 91YF82M8AA]
sources:
  - url: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)
    locator: Darboux function
---

## Definition

Let $I \subseteq \R$ be an [[interval|0TK8Y3J5RW]] and let $f \colon I \to \R$ be a [[function|0XHDTE5YMZ]].
The function $f$ is a *Darboux function*, or has the *intermediate value property*, if the following holds.
For all $a < b$ in $I$ and each real $y$ between $f(a)$ and $f(b)$, there is $c \in [a, b]$ with $f(c) = y$.

## Remarks

Equivalently, $f$ maps every subinterval of $I$ onto an interval.
Each [[continuous function|0P27W9RW0G]] on $I$ is a Darboux function, by the [[intermediate value theorem|00KAFBR32E]].
Each derivative on an open interval is a Darboux function, by [[Darboux's theorem|0Q5A47J48D]].

Example: let $f(x) = \sin(1/x)$ for $x \ne 0$ and $f(0) = 0$, on $\R$.
On a subinterval not containing $0$, $f$ is continuous, so its image is an interval.
A subinterval with more than one point that contains $0$ contains a full period of $\sin(1/x)$ on one side of $0$, so its image is $[-1, 1]$.
So $f$ is a Darboux function, although it is not continuous at $0$.

Non-example: the floor function $\floor{x}$ on $\R$ has $\floor{0} = 0$ and $\floor{1} = 1$, but it never takes the value $1/2$.

The [[Conway base 13 function|91YF82M8AA]] is a Darboux function that is continuous at no point.