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Some Remarks on Extreme Derivates

Published online by Cambridge University Press:  20 November 2018

A. M. Bruckner
A. M. Bruckner*
Affiliation:
University of California, Santa Barbara
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In 1957 Hájek [1] proved that the extreme bilateral derivates of an arbitrary finite real valued function of a real variable, are Borel measurable of class ≦ 2. It was later shown by Staniszewska [3] that Hájek's result is the best possible (even among the class of functions satisfying a Lipschitz condition). Staniszewska exhibited a Eipschitz function whose extreme bilateral derivates are not in Borel class 1. Staniszewska's proof makes use of a result of Zahorski's [4] concerning kernel functions.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 4 , August 1969 , pp. 385 - 388
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Hájek, O.,Note sur la mesurabilité B de la dérivée supérieure. Fund. Math. 44 (1957) 238–240.Google Scholar
2. Morse, A., Dini derivatives of continuous functions. Proc. Amer. Math. Soc. 5 (1954) 126–130.Google Scholar
3. Staniszewska, J., Sur la class de Baire des dérivées de Dini. Fund. Math. 47 (1959) 215–217.Google Scholar
4. Zahorski, Z., Sur les ensembles des points de divergence de certains intégrales singulières. Ann. Soc. Pol. Math. 19 (1946) 66–105.Google Scholar