Differentiable on an open set

Builds on Differentiable function of several variables

Definition

Let $U \subseteq \mathbb{R}^n$ be an open set and let $f \colon U \to \mathbb{R}^m$ . The function $f$ is differentiable on $U$ when it is differentiable at each point $a \in U$ . Then the derivative of $f$ is the map $Df \colon U \to L(\mathbb{R}^n, \mathbb{R}^m)$ , $a \mapsto Df(a)$ , where $L(\mathbb{R}^n, \mathbb{R}^m)$ is the space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ .

Remarks

The function $f$ is continuously differentiable, or of class $C^1$ , on $U$ when it is differentiable on $U$ and $Df$ is continuous. Here $L(\mathbb{R}^n, \mathbb{R}^m)$ carries the topology of $m \times n$ matrices. For a set $S$ that is not open, one uses derivatives within $S$ instead. On an open set the two notions agree. Each polynomial map $\mathbb{R}^n \to \mathbb{R}^m$ is differentiable on $\mathbb{R}^n$ .

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Euclidean space object
Function concept
Function of several variables concept
Limit of a function of several variables concept
Differentiable function of several variables definition
Differentiable on an open set definition

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Tag	`08S7FDRS5Z`
URI	`m2://08S7FDRS5Z`
Kind	definition
Writing	W0 stub
MSC	26B05
Defines	Differentiable function of several variables
Uses	Differentiable function of several variables

Differentiable on an open set

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