a1 Department of Mathematical Sciences, University of Petroleum and Minerals, UPM Box 639, Dhahran 31261, Saudi Arabia
We say that a function has locally small Riemann sums on an interval if for each point x in the interval, and for each positive number ε, all sufficiently fine partitions of intervals lying in neighborhood of x but not containing x have Riemann sums of absolute value less than ε. The main result is then as the title states. We use the generalized Riemann approach to Perron integration, assuming that functions are measurable only to insure that conditions involving the positive and negative parts of the functions are satisfied.
(Received June 05 1984)
1980 Mathematics subject classification (Amer. Math. Soc)