The equality of unilateral derivates
Authors:
M. J. Evans and P. D. Humke
Journal:
Proc. Amer. Math. Soc. 79 (1980), 609613
MSC:
Primary 26A24; Secondary 26A45
MathSciNet review:
572313
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Abstract: C. J. Neugebauer has shown that if f is a continuous function of bounded variation defined on the real line, then the set E where the upper right derivate differs from the upper left derivate is of measure zero and first category. Here it is shown that this result is best possible; that is, given any measure zero first category set K, there is a continuous function of bounded variation for which . It is also shown that if f is monotone, then E is porous. This result can be used to provide an elementary proof of the fact that for an arbitrary function f the left and right essential cluster sets are identical except at a porous set of points, a result first proved by L. Zajíček.
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C.
L. Belna, M.
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 [1]
 C. L. Belna, M. J. Evans and P. D. Humke, Symmetric and ordinary differentiation, Proc. Amer. Math. Soc. 72 (1978), 261267. MR 507319 (80d:26006)
 [2]
 E. P. Dolženko, Boundary properties of arbitrary functions, Math. USSRIzv. 1 (1967), 112.
 [3]
 M. Kulbacka, Sur l'ensemble des points de l'asymetrie approximative, Acta Sci. Math. (Szeged) 21 (1960), 9095. MR 0117307 (22:8088)
 [4]
 L. Mišík, Über approximative derivierte Zahlen monotoner Funktionen, Czechoslovak Math. J. 26 (101) (1976), 579583. MR 0432834 (55:5814)
 [5]
 C. J. Neugebauer, A theorem on derivates. Acta Sci. Math. (Szeged) 23 (1962), 7981. MR 0140624 (25:4041)
 [6]
 L. Zajíček, On cluster sets of arbitrary functions, Fund. Math. 83 (1974), 197217. MR 0338294 (49:3060)
 [7]
 , Sets of porosity and sets of porosity (q), Časopis Pěst. Mat. 101 (1976), 350359. MR 0457731 (56:15935)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198005723131
PII:
S 00029939(1980)05723131
Keywords:
Derivates,
bounded variation,
monotone functions,
porosity
Article copyright:
© Copyright 1980
American Mathematical Society
