Qualitative differentiation

Evans, Michael J.; Larson, Lee

doi:10.1090/S0002-9947-1983-0712262-6

Qualitative differentiation
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by Michael J. Evans and Lee Larson

Trans. Amer. Math. Soc. 280 (1983), 303-320

DOI: https://doi.org/10.1090/S0002-9947-1983-0712262-6

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Abstract:

Qualitative derivates and derivatives, as well as qualitative symmetric derivates and derivatives, are studied in the paper. Analogues of several results known for ordinary derivates and derivatives are obtained in the qualitative setting.

References

C. L. Belna, G. T. Cargo, M. J. Evans, and P. D. Humke, Analogues of the Denjoy-Young-Saks theorem, Trans. Amer. Math. Soc. 271 (1982), no. 1, 253–260. MR 648091, DOI 10.1090/S0002-9947-1982-0648091-0
C. L. Belna, M. J. Evans, and P. D. Humke, Symmetric and ordinary differentiation, Proc. Amer. Math. Soc. 72 (1978), no. 2, 261–267. MR 507319, DOI 10.1090/S0002-9939-1978-0507319-2
C. L. Belna, M. J. Evans, and P. D. Humke, Symmetric monotonicity, Acta Math. Acad. Sci. Hungar. 34 (1979), no. 1-2, 17–22. MR 546713, DOI 10.1007/BF01902587

The derivates and approximate derivates of measurable functions

32

Shu-Er Chow, On approximate derivatives, Bull. Amer. Math. Soc. 54 (1948), 793–802. MR 26114, DOI 10.1090/S0002-9904-1948-09082-6

Boundary properties of arbitrary functions

1

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

Recherches sur la structure des fonctions mesurables

31

Recherches sur la structure des fonctions mesurables

9

Lee Larson, The Baire class of approximate symmetric derivates, Proc. Amer. Math. Soc. 87 (1983), no. 1, 125–130. MR 677247, DOI 10.1090/S0002-9939-1983-0677247-2
Lee Larson, The symmetric derivative, Trans. Amer. Math. Soc. 277 (1983), no. 2, 589–599. MR 694378, DOI 10.1090/S0002-9947-1983-0694378-6
John L. Leonard, Some conditions implying the monotonicity of a real function, Rev. Roumaine Math. Pures Appl. 17 (1972), 757–780. MR 304574
S. Marcus, La limite approximative qualitative, Com. Acad. R. P. Romîne 3 (1953), 9–12 (Romanian, with French and Russian summaries). MR 70686
S. Marcus, La continuité approximative qualitative, Com. Acad. R. P. Romîne 3 (1953), 117–120 (Romanian, with French and Russian summaries). MR 70687

Sur la dérivée approximative qualitative

C. J. Neugebauer, A theorem on derivates, Acta Sci. Math. (Szeged) 23 (1962), 79–81. MR 140624
John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443
Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
B. S. Thomson, On full covering properties, Real Anal. Exchange 6 (1980/81), no. 1, 77–93. MR 606543
Luděk Zajíček, Sets of $\sigma$-porosity and sets of $\sigma$-porosity $(q)$, Časopis Pěst. Mat. 101 (1976), no. 4, 350–359 (English, with Russian summary). MR 0457731
L. Zajíček, On the symmetry of Dini derivates of arbitrary functions, Comment. Math. Univ. Carolin. 22 (1981), no. 1, 195–209. MR 609947
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776