Real functions Functions of several real variables Theorem

Whitney extension theorem

Builds on Closed set Multi-index notation C^k function characterised by continuous derivatives Taylor's theorem in several variables

Statement

Let $n \ge 1$ and $m \ge 0$ be integers and let $A \subseteq \R^n$ be a closed set. For each multi-index $\alpha$ with $\abs{\alpha} \le m$ , let $f_\alpha \colon A \to \R$ be continuous. For $x, y \in A$ and $\abs{\alpha} \le m$ , set $R_\alpha(x, y) = f_\alpha(x) - \sum_{\abs{\beta} \le m - \abs{\alpha}} \frac{f_{\alpha + \beta}(y)}{\beta!} (x - y)^\beta.$ Assume that for every compact set $K \subseteq A$ and every $\abs{\alpha} \le m$ , $\frac{\abs{R_\alpha(x, y)}}{\norm{x - y}^{m - \abs{\alpha}}} \to 0 \quad \text{as } \norm{x - y} \to 0,$ uniformly for $x, y \in K$ with $x \ne y$ . Then there is a function $F \colon \R^n \to \R$ of class $C^m$ such that $D^\alpha F = f_\alpha$ on $A$ for every $\abs{\alpha} \le m$ . Moreover, $F$ can be chosen of class $C^\infty$ on $\R^n \setminus A$ .

Remarks

The family $(f_\alpha)$ is a list of candidate derivatives of $F$ on $A$ , called a jet. The condition says that the candidate values obey Taylor's theorem uniformly on compact parts of $A$ . It is necessary: if $F$ is of class $C^m$ on $\R^n$ , then the restrictions of $D^\alpha F$ to $A$ satisfy it, by Taylor's theorem. For $m = 0$ the condition holds automatically, because a continuous function is uniformly continuous on compact sets, and the theorem gives a continuous extension, as the Tietze extension theorem does. The proof covers $\R^n \setminus A$ by cubes whose size is comparable to their distance to $A$ . It then glues Taylor polynomials taken at nearby points of $A$ with a partition of unity subordinate to these cubes.

Going deeper

For a student

The theorem turns questions about functions on arbitrary closed sets into questions about jets, which are easier to handle. It is used to define smooth functions on closed subsets, and in the study of ideals of smooth functions. For $n = 1$ and $m = 1$ the condition says that $f_0$ is differentiable along $A$ with derivative $f_1$ , uniformly on compact sets. The related Borel's lemma extends a jet given at a single point.

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Closed set concept
Euclidean space object
Function concept
Fréchet derivative definition
Function of several variables concept
Limit of a function of several variables concept
Differentiable function of several variables definition
Partial derivative definition
Differentiable on an open set definition
Higher partial derivatives definition
Iterated derivative definition
Multi-index notation concept
C^k function characterised by continuous derivatives statement
Taylor's theorem in several variables statement
Whitney extension theorem statement

Tag	`RH2EZGYGAW`
URI	`m2://RH2EZGYGAW`
Kind	statement, theorem
Writing	W0 stub
MSC	26B05, 26E10, 54C20
Status	This copy of the site has no build data. Run `m2 status RH2EZGYGAW`.
About	C^k function characterised by continuous derivatives
Uses	Closed set, Multi-index notation, C^k function characterised by continuous derivatives, Taylor's theorem in several variables
Same as	Wikidata Q7996766 Wikipedia Whitney extension theorem
Changed	2 November 2026