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Lipschitz restrictions of continuous functions and a simple construction of Ulam-Zahorski \(C^1\) interpolation. (English) Zbl 1404.26006

S. Agronsky et al. [Trans. Am. Math. Soc. 289, 659–677 (1985; Zbl 0601.26007)] proved the following:
Theorem. For every continuous function \(f:\mathbb{R}\rightarrow \mathbb{R}\) there exists a continuously differentiable function \(g:\mathbb{R}\rightarrow \mathbb{R}\) such that the set \(f=g\) is uncountable.
In this paper, an elementary proof of this theorem is given. Namely, it is done using a simple argument according to which for every continuous function \(f:\mathbb{R}\rightarrow \mathbb{R}\) its restriction to some perfect set is Lipschitz.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26B05 Continuity and differentiation questions

Citations:

Zbl 0601.26007
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