Abstract
In this paper, we introduced a Henstock-type integral named N-integral of a real valued function f on a closed and bounded interval [a,b]. The set N-integrable functions lie entirely between Riemann integrable functions and Henstock-Kurzweil integrable functions. Furthermore, this new integral integrates all improper Riemann integrable functions even if they are not Lebesgue integrable. It was shown that for a Henstock-Kurzweil integrable function f on [a,b], the following are equivalent:The function f is N-integrable;There exists a null set S for which given epsilon >0 there exists a gauge delta such that for any delta-fine partial division D={(xi,[u,v])} of [a,b] we have [(phi_S(D) Gamma_epsilon) sum |f(v)-f(u)||v-u|<epsilon] where phi_S(D)={(xi,[u,v])in D:xi not in S} and [Gamma_epsilon={(xi,[u,v]): |f(v)-f(u)|<= epsilon}] andThe function f is continuous almost everywhere. A characterization of continuous almost everywhere functions was also given.
Highlights
INTRODUCTIONThe idea is to exclude some problematic point-interval pairs of a division in forming the Riemann sum
Recall that a real valued function f on [a, b] is said to be Riemann integrable to A if for every > 0 there exists a constant δ > 0 such that for any division D of [a, b] given by a = x0 < x1 < · · · < xn = b and {ξ1, ξ2, . . . , ξn}2020 Mathematics Subject Classification: 26A39, 26A42 Received: 04-08-2019, accepted: 27-02-2020.with xi−1 ≤ ξi ≤ xi and xi − xi−1 < δ for all i, we have n f(xi − xi−1) − A < . (1) i=1One may see [1], [4], [6], or [8] for further details
The whole paper is composed of five sections
Summary
The idea is to exclude some problematic point-interval pairs of a division in forming the Riemann sum. This new integral can integrate all improper Riemann integrable functions and even more. Characterizations of the N -integrable functions will be given by looking at the points of discontinuity. The development of the characterizations utilizes the idea of Γ which was adopted from [2] and [9] This time Γ is defined based on discontinuity instead of nondifferentiability. We explicitly identify the set that would optimally identify the point-interval pairs that are to be excluded in forming the Riemann sums. In the last section, we present examples to strengthen our results
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