Continuous partial derivatives imply differentiability

Builds on Partial derivative Differentiable function of several variables

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ , and let $a \in U$ . Suppose that each partial derivative $\partial f_j / \partial x_i$ exists at each point of $U$ , for $1 \le i \le n$ and $1 \le j \le m$ , and that each of them is continuous at $a$ . Then $f$ is differentiable at $a$ .

Remarks

If the partial derivatives exist on $U$ and are continuous on all of $U$ , then $f$ is differentiable on $U$ , and $Df$ is continuous. So $f$ is of class $C^1$ on $U$ . The converse is false. The function $f(x) = \|x\|^2 \sin(1/\|x\|)$ for $x \ne 0$ , with $f(0) = 0$ , is differentiable on $\mathbb{R}^n$ , but its partial derivatives are not continuous at $0$ . Without continuity, the existence of the partial derivatives is not enough, see this counterexample.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Euclidean space object
Function concept
Function of several variables concept
Limit of a function of several variables concept
Partial derivative concept
Differentiable function of several variables definition
Continuous partial derivatives imply differentiability statement

Tag	`0KXSK8TEEB`
URI	`m2://0KXSK8TEEB`
Kind	statement, theorem
Writing	W0 stub
MSC	26B05
Status	This copy of the site has no build data. Run `m2 status 0KXSK8TEEB`.
Uses	Partial derivative, Differentiable function of several variables

Continuous partial derivatives imply differentiability

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here