---
tag: 07J5X2T1ZX
kind: statement
role: corollary
name: A function with positive derivative is strictly increasing
msc: [26A24]
uses: [0BARQY52N5, 0TR9SR9FTB]
sources:
  - url: https://en.wikipedia.org/wiki/Monotonic_function
    locator: Some basic applications and results
---

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous, and differentiable at each interior point of $I$.
If $f'(x) > 0$ for each interior point $x$ of $I$, then $f$ is [[strictly increasing|0TR9SR9FTB]] on $I$.

## Remarks

For $x < y$ in $I$, the [[mean value theorem|0BARQY52N5]] gives $f(y) - f(x) = f'(c)(y - x) > 0$.
The converse fails: $x^3$ is strictly increasing, but its derivative at $0$ is $0$.
Mathlib states it as `strictMonoOn_of_deriv_pos`.

## Explanations

### L1 human

If the graph has positive slope everywhere, it rises from left to right.