Symmetric semicontinuity implies continuity
HTML articles powered by AMS MathViewer
- by Jaromir Uher PDF
- Trans. Amer. Math. Soc. 293 (1986), 421-429 Request permission
Abstract:
The main result of this paper shows that, for any function, symmetric semicontinuity on a measurable set $E$ implies continuity a.e. in $E$ and, similarly, that symmetric semicontinuity on a set residual in $R$ implies continuity on a set residual in $R$. These propositions are used to prove more precise versions of the fundamental connections between symmetric and ordinary differentiability.References
- C. L. Belna, Symmetric continuity of real functions, Proc. Amer. Math. Soc. 87 (1983), no. 1, 99–102. MR 677241, DOI 10.1090/S0002-9939-1983-0677241-1 Z. Charzynski, Sur les fonctions dont la derivée symetrique est partout finie, Fund. Math. 21 (1933), 214-225. H. Fried, Über die symmetrische Stetigkeit von Funktionen, Fund. Math. 29 (1937), 134-137. A. Khintchine, Recherches sur la structure des fonctions mesurables, Fund. Math. 9 (1927), 212-279.
- S. P. Ponomarev, Symmetrically continuous functions, Mat. Zametki 1 (1967), 385–390 (Russian). MR 209412
- David Preiss, A note on symmetrically continuous functions, Časopis Pěst. Mat. 96 (1971), 262–264, 300 (English, with Czech summary). MR 0306411 S. Saks, Théorie de l’intégrale, PWN, Warsaw, 1933.
- E. M. Stein and A. Zygmund, On the differentiability of functions, Studia Math. 23 (1963/64), 247–283. MR 158955, DOI 10.4064/sm-23-3-247-283
- Jaromír Uher, Symmetrically differentiable functions are differentiable almost everywhere, Real Anal. Exchange 8 (1982/83), no. 1, 253–261. MR 694513 W. H. Young, La symétrie de structure des fonctions de variables réelles, Bull. Sci. Math. 52 (1928), 265-280.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 293 (1986), 421-429
- MSC: Primary 26A15; Secondary 26A24
- DOI: https://doi.org/10.1090/S0002-9947-1986-0814930-7
- MathSciNet review: 814930