We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.
@article{bwmeta1.element.bwnjournal-article-fmv141i1p85bwm,
author = {Jack Brown},
title = {Continuous-, derivative-, and differentiable-restrictions of measurable functions},
journal = {Fundamenta Mathematicae},
volume = {141},
year = {1992},
pages = {85-95},
zbl = {0810.28003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv141i1p85bwm}
}
Brown, Jack. Continuous-, derivative-, and differentiable-restrictions of measurable functions. Fundamenta Mathematicae, Tome 141 (1992) pp. 85-95. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv141i1p85bwm/
[00000] [1] S. Agronsky, A. M. Bruckner, M. Laczkovich and D. Preiss, Convexity conditions and intersections with smooth functions, Trans. Amer. Math. Soc. 289 (1985), 659-677. | Zbl 0601.26007
[00001] [2] H. Blumberg, New properties of all real functions, ibid. 24 (1922), 113-128.
[00002] [3] J. B. Brown, Differentiable restrictions of real functions, Proc. Amer. Math. Soc. 108 (1990), 391-398. | Zbl 0692.26002
[00003] [4] J. B. Brown and K. Prikry, Variations on Lusin's theorem, Trans. Amer. Math. Soc. 302 (1987), 77-86. | Zbl 0619.28005
[00004] [5] A. M. Bruckner, J. G. Ceder and M. L. Weiss, On the differentiability structure of real functions, ibid. 142 (1969), 1-13. | Zbl 0182.38301
[00005] [6] J. Ceder, Some examples on continuous restrictions, Real Anal. Exchange 7 (1981/ 82), 155-162. | Zbl 0533.26002
[00006] [7] F. Filipczak, Sur les fonctions continues relativement monotones, Fund. Math. 58 (1966), 75-87. | Zbl 0185.12204
[00007] [8] V. Jarník, Sur les nombres dérivés approximatifs, ibid. 22 (1934), 4-16.
[00008] [9] C. Kuratowski, La propriété de Baire dans les espaces métriques, ibid. 16 (1930), 390-394. | Zbl 56.0846.03
[00009] [10] K. Kuratowski and A. Mostowski, Set Theory with an Introduction to Descriptive Set Theory, North-Holland, Amsterdam 1976. | Zbl 0337.02034
[00010] [11] M. Laczkovich, Differentiable restrictions of continuous functions, Acta Math. Hungar. 44 (1984), 355-360. | Zbl 0558.26005
[00011] [12] N. Lusin, Sur les propriétés des fonctions mesurables, C. R. Acad. Sci. Paris 154 (1912), 1688-1690. | Zbl 43.0484.04
[00012] [13] N. Lusin, Sur la recherche des fonctions primitives, ibid. 162 (1916), 975-978. | Zbl 46.0390.02
[00013] [14] E. Marczewski (Szpilrajn), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1935), 17-34. | Zbl 61.0229.01