---
tag: 0A61B60B5R
kind: definition
name: Differentiable function
msc: [26A24]
defines: 0K57YG3BB8
uses: [0F1WR9C83K]
sources:
  - url: https://en.wikipedia.org/wiki/Differentiable_function
    locator: Differentiability of real functions of one variable
---

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is differentiable at $a \in I$ if $f$ has a [[derivative|0F1WR9C83K]] at $a$.
It is differentiable on $I$ if it is differentiable at each point of $I$.

## Remarks

[[Differentiable functions are continuous|07ZMX4ANJS]].
The converse fails: the [[absolute value|0CQ9ZM383N]] is continuous but not differentiable at $0$.
Polynomials, the exponential function, and the sine are differentiable on $\mathbb{R}$.
Mathlib defines differentiability at a point as `DifferentiableAt`.

## Explanations

### L1 human

A differentiable function has a well defined slope at each point.
Its graph is smooth, with no corners and no vertical tangents.