A symmetric density property: monotonicity and the approximate symmetric derivative

Authors:
C. Freiling and D. Rinne

Journal:
Proc. Amer. Math. Soc. **104** (1988), 1098-1102

MSC:
Primary 26A48; Secondary 26A24

DOI:
https://doi.org/10.1090/S0002-9939-1988-0936773-3

MathSciNet review:
936773

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Abstract | References | Similar Articles | Additional Information

Abstract: The following is established:

Let and be open sets of real numbers whose union has full measure. If for each , the set has density zero at zero, then these sets are all empty.

This is then used to prove the following:

If is a continuous real valued function with a nonnegative lower approximate symmetric derivative, then is nondecreasing.

**[1]**N. K. Kundu,*On the approximate symmetric derivative*, Colloq. Math.**28**(1973), 275-285. MR**0327991 (48:6333)****[2]**L. Larson,*Symmetric real analysis: A survey*, Real Anal. Exchange**9**(1983-1984), 154-178. MR**742782 (85f:26024)****[3]**-,*Monotonicity and the approximate symmetric derivative*, Real Anal. Exchange**12**(1986-1987), 121-123.**[4]**J. Matousek,*Approximate symmetric derivatives and monotonicity*, Comment Math. Univ. Carolin.**27**(1986), 83-86. MR**843422 (87f:26004)****[5]**S. N. Mukhopadhyay,*On approximate Schwarz differentiability*, Monatsh. Math.**70**(1966), 454-460. MR**0202937 (34:2796)****[6]**H. W. Pu and H. H. Pu,*On approximate Schwarz derivatives*, Rev. Roumaine Math. Pures Appl.**25**(1980), 257-264. MR**577036 (81g:26003)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0936773-3

Keywords:
Symmetric derivative,
approximate symmetric derivative,
density,
monotonicity

Article copyright:
© Copyright 1988
American Mathematical Society