Upper and Lower N-integrals - Cabral, Emmanuel A (Archīum Ateneo, 2019)
. This paper provides alternative definitions of the N−integral using the set of discontinuity Df of the function f . Upper and lower Darboux sums are introduced so that a Darboux characterization of the N−integral similar to the Darboux definition of the Riemann integral is obtained. It is also shown that a function is N− integrable with integral A if and only if for every > 0, there exists an elementary set E with [a, b] \ E of measure smaller than and S ∞ ⊂ [a, b] \ E such that f is Riemann integrable on E and (R) E f − A < . Here S ∞ is the set of all points in [a, b] such that for every x ∈ S ∞, there exists a sequence {xn} in [a,b] with | f(xn)|→∞ as n → ∞