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On the extension of differentiable functions. (English) Zbl 0063.08236
From the introduction: The author has shown previously how to extend the definition of a function of class $$C^m$$ defined in a closed set $$A$$ so it will be of class $$C^m$$ throughout space (see (*) [Trans. Am. Math. Soc. 36, 63–89 (1934; Zbl 0008.24902)]). Here we shall prove a uniformity property: If the function and its derivatives are sufficiently small in $$A$$, then they may be made small throughout space. Besides being bounded, we assume that $$A$$ has the following property:{
}(P) There is a number $$\omega$$ such that any two points $$x$$ and $$y$$ of $$A$$ are joined by an arc in $$A$$ of length less than or equal to $$\omega r_{xy}$$ $$(r_{xy}$$ being the distance between $$x$$ and $$y)$$.{
}This property was made use of in [Ann. Math. (2) 35, 482–485 (1934; Zbl 0009.30901)]; its necessity in the theorem is shown by two examples below.{
}A second theorem removes the boundedness condition in the first theorem, and weakens the hypothesis (P); its proof makes use of the proof of the first theorem. We remark that in each theorem, as in [(*) ], the extended function is a linear functional of its values in $$A$$.{
}The proof of Theorem 1 is obtained by examining the proof in [(*)];

##### MSC:
 26-XX Real functions
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