Analyticity of almost everywhere differentiable functions
Author:
Eric J. Howard
Journal:
Proc. Amer. Math. Soc. 110 (1990), 745753
MSC:
Primary 26B05; Secondary 26E05, 30B40
MathSciNet review:
1027093
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We develop a partitioning lemma (see Lemma 5) for superadditive set functions satisfying certain continuity conditions. This leads to a relatively simple proof of two theorems of A. S. Besicovitch on when a function of a complex variable that is continuous and differentiable outside of small exceptional sets is analytic (or almost everywhere equal to an analytic function).
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Ralph
Henstock, Lectures on the theory of integration, Series in
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1988. MR
963249 (91a:28001)
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F. Pfeffer, A note on the lower derivate of a set
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F. Pfeffer, On the lower derivate of a set function, Canad. J.
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Washek
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F. Pfeffer and W.
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(38 #6529), http://dx.doi.org/10.1090/S00029939196902382538
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L. Royden, A generalization of Morera’s theorem, Ann.
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 [B]
 A. S. Besicovitch, On sufficient conditions for a function to be analytic, and behaviour of analytic functions in the neighborhood of nonisolated singular points, Proc. London Math. Soc. 32 (1931), 19.
 [Fa]
 K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, Cambridge, 1985. MR 867284 (88d:28001)
 [Fe]
 H. Federer, Geometric measure theory, SpringerVerlag, New York, 1969. MR 0257325 (41:1976)
 [H]
 R. Henstock, Lectures on the theory of integration, World Scientific, Singapore, 1988. MR 963249 (91a:28001)
 [P1]
 W. F. Pfeffer, A note on the lower derivate of a set function and semihereditary systems of sets, Proc. Amer. Math. Soc. 18 (1967), 10201025. MR 0218503 (36:1589)
 [P2]
 , On the lower derivate of a set function, Canad. J. Math. 20 (1968), 14891498. MR 0232906 (38:1229)
 [P3]
 , The multidimensional fundamental theorem of calculus, J. Australian Math. Soc. 43 (1987), 143170. MR 896622 (89b:26013)
 [P4]
 , Divergence theorem for vector fields with singularities, unpublished.
 [P5]
 , Stokes theorem for forms with singularities, C. R. Acad. Sci. Paris, Sér. I 306 (1988), 589592. MR 941632 (89k:26008)
 [PW]
 W. F. Pfeffer and W. J. Wilber, A note on cluster points of a semihereditary stable system of sets, Proc. Amer. Math. Soc. 21 (1969), 121125. MR 0238253 (38:6529)
 [PY]
 W. F. Pfeffer and WeiChi Yang, A multidimensional variational integral and its extensions, Real Analysis Exchange (1) 15 (198990), 111169. MR 1042534 (91b:26016)
 [R]
 H. L. Royden, A generalization of Morera 's theorem, Ann. Polon. Math. 12 (1962), 199202. MR 0141766 (25:5163)
 [S]
 S. Saks, Theory of the integral, Dover, New York, 1964. MR 0167578 (29:4850)
 [Z]
 L. Zalcman, Analyticity and the Pompeiu problem, Arch. Rational Mech. Anal. 47 (1972), 237254. MR 0348084 (50:582)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010270939
PII:
S 00029939(1990)10270939
Keywords:
fine partition,
exceptional set,
lower derivate
Article copyright:
© Copyright 1990 American Mathematical Society
