## Pointwise differentiability and absolute continuity

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- by Thomas Bagby and William P. Ziemer
- Trans. Amer. Math. Soc.
**191**(1974), 129-148 - DOI: https://doi.org/10.1090/S0002-9947-1974-0344390-6
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## Abstract:

This paper is concerned with the relationships between ${L_p}$ differentiability and Sobolev functions. It is shown that if*f*is a Sobolev function with weak derivatives up to order

*k*in ${L_p}$, and $0 \leq l \leq k$, then

*f*has an ${L_p}$ derivative of order

*l*everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an ${L_p}$ derivative everywhere except for a set small in capacity and if these derivatives are in ${L_p}$, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.

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## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**191**(1974), 129-148 - MSC: Primary 26A54; Secondary 46E35
- DOI: https://doi.org/10.1090/S0002-9947-1974-0344390-6
- MathSciNet review: 0344390