If we have a finitely many discontinuity points on an interval, say [a,b] and we want to compute the integral of a function defined on this interval, usually we use the notion of improper integral and split the integral in a sum of integral and then compute each sum as an improper integral itself, but if we wave infinetely many points of discontinuity, what can be done?
Thank you.
Even if uncountable, the real question is "what is the measure of the set of discontinuities"? (Any countable set has measure 0, an uncountable set may have measure 0.) Any set of measure 0 can be ignored.