---
tag: 0Q5A47J48D
kind: statement
role: theorem
name: Darboux's theorem
aliases: [intermediate-value-theorem-for-derivatives]
msc: [26A24]
uses: [0WS1A8E2P9, 0PCVBRV9VJ, 0HXEX0DPRA]
sources:
  - url: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)
    locator: Statement of the theorem
same_as:
  wikidata: Q660799
  wikipedia: Darboux's theorem (analysis)
---

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be differentiable on $[a, b]$, with [[one-sided derivatives|0HXEX0DPRA]] at the endpoints.
Let $y$ be a real number strictly between $f'(a)$ and $f'(b)$.
Then there is $c \in (a, b)$ with $f'(c) = y$.

## Remarks

Assume $f'(a) < y < f'(b)$. The function $g(x) = f(x) - y x$ has $g'(a) < 0 < g'(b)$. So its minimum on $[a, b]$, given by the [[extreme value theorem|0WS1A8E2P9]], is at an interior point $c$. [[Fermat's theorem|0PCVBRV9VJ]] gives $g'(c) = 0$.
So a derivative has the intermediate value property, even when it is not continuous.
In particular, a derivative has no jump discontinuities. Mathlib states it as `exists_hasDerivWithinAt_eq_of_gt_of_lt`.

## Explanations

### L1 human

The derivative of a function cannot skip values, just as a continuous function cannot.
For example, no function has derivative $0$ on the left of a point and $1$ on the right.