---
tag: 07BY47HG5S
kind: statement
role: theorem
name: Continuity of the inverse of a strictly monotone continuous function
msc: [26A48]
uses: [0TR9SR9FTB, 00KAFBR32E, 0KJXHN0TW0]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_function
    locator: Standard inverse functions
---

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous and [[strictly increasing|0TR9SR9FTB]].
Then $J = f(I)$ is an interval, and the inverse function $f^{-1} \colon J \to I$ is continuous and strictly increasing.

## Remarks

The set $J$ is an interval by the [[intermediate value theorem|00KAFBR32E]].
The inverse is strictly increasing and its image is the interval $I$. A monotone function whose image is an interval has no jumps, so $f^{-1}$ is continuous.
The same holds for strictly decreasing functions.
This gives the continuity of $n$-th roots, of the [[logarithm|0044CPHJBV]], and of the inverse trigonometric functions.

## Explanations

### L1 human

The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.
A rising unbroken graph stays rising and unbroken after the reflection.