---
tag: 0135XZZQ8M
kind: definition
name: Differentiable function of several variables
msc: [26B05]
defines: 0HHHDCXZV4
uses: [0XPR1E85W0, 0J8Y1NCY0K]
sources:
  - url: https://en.wikipedia.org/wiki/Differentiable_function
    locator: Differentiability in higher dimensions
  - url: https://en.wikipedia.org/wiki/Derivative_(multivariable_calculus)
    locator: Differentiability
---

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
The function $f$ is differentiable at $a$ when there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that
$$\lim_{h \to 0} \frac{\|f(a + h) - f(a) - L h\|}{\|h\|} = 0.$$
Here the limit is over $h \ne 0$ with $a + h \in U$, and $\|\cdot\|$ is the Euclidean norm.
The map $L$ is then unique. It is the derivative $Df(a)$, also called the [[total derivative|0FNKFT8H6E]] of $f$ at $a$.

## Remarks

This is the [[Fréchet derivative|08EVNYZF42]] for the spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where each linear map is continuous.
A function that is differentiable at $a$ is continuous at $a$, and all its [[directional derivatives|06KWGHDN2J]] exist at $a$.
The converse fails: a function can have all partial derivatives at a point and not be differentiable there, see [[this counterexample|0TNZC55X3J]].
For $n = m = 1$, the definition is the usual one, with $L h = f'(a) h$.

## Explanations

### L1 human

A differentiable function looks flat when you zoom in on its graph near the point.
For $f(x, y) = x^2 + y^2$ at $(1, 1)$, the change is $f(1 + h_1, 1 + h_2) - f(1, 1) = 2 h_1 + 2 h_2 + h_1^2 + h_2^2$.
The linear part is $L h = 2 h_1 + 2 h_2$, and the rest $h_1^2 + h_2^2 = \|h\|^2$ is small compared with $\|h\|$.