Basically from what I understand the integral of a function, say ∫x^2dx from say 0 to 1, can be represented as the supremum/infimum of the function values within each of a countably infinite number of vanishingly small intervals in the domain created by a countably infinite number of partition points, each multiplied against the corresponding length of the interval within which each supremum/infimum resides.(adsbygoogle = window.adsbygoogle || []).push({});

By The Nested Interval Theorem (I think!) we know each of these intervals has only a single value as n->∞, which means that the infimum=supremum, the upper and lower integrals are equal, thus the integral exists etc.

My concern is that each of these intervals has length 1/n as n->∞, which means they can be put into a one-to-one correspondence with the natural numbers, meaning the number of heights at which the integral is evaluated compose a countably infinite set. Yet! the domain of the original function is the real numbers, which are uncountably infinite! I'm still a novice in the areas of higher mathematics but to my mathematical intuition this implies that we are using methods that are only precise to the level of a countably infinite set to approximate the area under the curve of a function which has an uncountably infinite domain. Does this not entail that we are not taking into account all possible inputs that the function can take? Is it not possible that this function is behaving in strange ways, possibly where L(ƒ,Pn)≠U(ƒ,Pn), if we were to zoom infinitely in on some point of the curve?

Hopefully some of you more experienced analysts can shed some light on this for me, and any other words of wisdom about analysis in general would be great to.

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# I am taking issue with the Riemann Integral

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