Vol. 47, No. 1, 1973

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
Peano derivatives and general integrals

Peter Southcott Bullen and S. N. Mukhopadhyay

Vol. 47 (1973), No. 1, 43–58

It is known that if a function f has a finite derivative (or approximate derivative) on a set E on which f is continuous then f is ACG (or ACG) on E and that if f is ACG (or ACG) on a set E then a finite derivative (or approximate derivative) of f exists almost everywhere in E. These results are extended by Sargent in the case of generalized derivatives of higher order. She has proved that if fn+1, the generalized derivative of f of order n + 1, exists in an interval [a,b] then the derivative fn is V n ACG on [a,b] and that if fn is V n ACG on [a,b] then fn+1 exists and is equal to the approximate derivative of fn almost everywhere in [a,b].

The present work is concerned with extending still further these results of Sargent by introducing a more general definitions of absolute continuity for the n-th derivatives. It also introduces an approximate Pn-integral which generalizes the Pn-integral of James and Bullen.

Mathematical Subject Classification 2000
Primary: 26A24
Received: 1 May 1972
Published: 1 July 1973
Peter Southcott Bullen
S. N. Mukhopadhyay