---
tag: 0P27W9RW0G
kind: definition
name: Continuous function
msc: [26A15]
defines: 0X3WGX95FE
uses: [0EEE2CZB2A]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_function
    locator: Definition
---

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is continuous if it is [[continuous at|0EEE2CZB2A]] each point of $D$.
For a subset $S \subseteq D$, $f$ is continuous on $S$ if its restriction to $S$ is continuous.

## Remarks

Sums, [[products|0VGMHCF4SC]], [[quotients|0GR2APWTAC]], and [[compositions|0NRQPQX3VA]] of continuous functions are continuous.
[[Polynomial functions are continuous|0C5TP6RYSV]]. So are the exponential function, the sine, and the cosine.
On a closed bounded interval, a continuous function has the [[intermediate value property|00KAFBR32E]] and [[attains its extreme values|0WS1A8E2P9]].
Mathlib defines it for topological spaces, as `Continuous`.

## Explanations

### L1 human

A continuous function has no jumps or breaks at any point of its domain.
On an interval, we can draw its graph without lifting the pen.