On the extension of differentiable functions.

*(English)*Zbl 0063.08236From 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 [(*)];

}(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 |