Analogues of the Denjoy-Young-Saks theorem

Belna, C. L.; Cargo, G. T.; Evans, M. J.; Humke, P. D.

doi:10.1090/S0002-9947-1982-0648091-0

Analogues of the Denjoy-Young-Saks theorem
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by C. L. Belna, G. T. Cargo, M. J. Evans and P. D. Humke PDF

Trans. Amer. Math. Soc. 271 (1982), 253-260 Request permission

Abstract:

In this paper, an analogue of the Denjoy-Young-Saks theorem concerning the almost everywhere classification of the Dini derivates of an arbitrary real function is established in both the case where the exceptional set is of first category and the case where it is $\sigma$-porous. Examples are given to indicate the sharpness of these results.

References

Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448
E. F. Collingwood, Cluster set theorems for arbitrary functions with applications to function theory, Ann. Acad. Sci. Fenn. Ser. A. I. 336(8) (1963), 15. MR 160914
E. P. Dolženko, Boundary value theorems on the uniqueness and behaviour of analytic functions near the boundary, Dokl. Akad. Nauk SSSR 129 (1959), 23–26 (Russian). MR 0107004

Boundary properties of arbitrary functions

1

Paul Erdős and George Piranian, Restricted cluster sets, Math. Nachr. 22 (1960), 155–158. MR 123719, DOI 10.1002/mana.19600220306
M. J. Evans and P. D. Humke, The equality of unilateral derivates, Proc. Amer. Math. Soc. 79 (1980), no. 4, 609–613. MR 572313, DOI 10.1090/S0002-9939-1980-0572313-1
Casper Goffman, On Lebesgue’s density theorem, Proc. Amer. Math. Soc. 1 (1950), 384–388. MR 36816, DOI 10.1090/S0002-9939-1950-0036816-1
I. P. Natanson, Theory of functions of a real variable. Vol. II, Frederick Ungar Publishing Co., New York, 1961. Translated from the Russian by Leo F. Boron. MR 0148805
C. J. Neugebauer, A theorem on derivates, Acta Sci. Math. (Szeged) 23 (1962), 79–81. MR 140624
Luděk Zajíček, Sets of $\sigma$-porosity and sets of $\sigma$-porosity $(q)$, Časopis Pěst. Mat. 101 (1976), no. 4, 350–359 (English, with Loose Russian summary). MR 0457731