Universally first return continuous functions

Authors:
Udayan B. Darji, Michael J. Evans and Richard J. O’Malley

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2677-2685

MSC:
Primary 26A21; Secondary 26A15, 26A24

DOI:
https://doi.org/10.1090/S0002-9939-1995-1233966-9

MathSciNet review:
1233966

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Abstract: It is known that the first return continuous functions are precisely the Darboux functions in Baire class 1, and that every such function can be changed via a homeomorphism into an approximately continuous function. Here we give two characterizations of the smaller class of universally first return continuous functions, one of which is the capacity of changing such a function via a homeomorphism into an approximately continuous function which is continuous almost everywhere.

**[1]**U. B. Darji, M. J. Evans, and R. J. O'Malley,*First return path systems: differentiability, continuity, and orderings*, Acta. Hungar. (to appear).**[2]**I. Maximoff,*Sur la transformation continue de quelques fonctions en derivees exactes*, Bull. Soc. Phys. Math. Kazan (3)**12**(1940), 57-81. MR**0015456 (7:420i)****[3]**R. J. O'Malley,*Approximately continuous functions which are continuous almost everywhere*, Acta Math. Acad. Sci. Hungar.**33**(1979), 395-402. MR**542489 (80g:26006)****[4]**-,*First return path derivatives*, Proc. Amer. Math. Soc.**116**(1992), 73-77. MR**1097349 (92k:26006)****[5]**D. Preiss,*Maximoff's Theorem*, Real Anal. Exchange**5**(1979), 92-104. MR**557966 (80m:26007)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1233966-9

Keywords:
First return continuity,
approximate continuity,
a.e. topology,
Baire class one,
Darboux

Article copyright:
© Copyright 1995
American Mathematical Society