Toward the concept of the definite integral and on the proof of the fundamental theorem of existence of integrals for ordinary differential equations
Abstract
The integrable functions also include those whose discontinuities can be enclosed within intervals, the sum of the lengths of which is arbitrarily small; indeed, by enclosing the points of discontinuity in intervals, we can represent the difference (b - a) as the sum of two parts, of which the first relates to the intervals lying, in turn, within the intervals enclosing the points of discontinuity (see the article). It is enough to show that the difference (b - a) can be made arbitrarily small only under one specific, well-defined law of division, one can then choose such division points that coincide with the endpoints of the intervals enclosing the discontinuities of the function f and then the difference (b - a) will indeed be arbitrarily small. From here we conclude that integrable functions include, inter alia, functions with a finite number of discontinuities, as well as functions whose points of discontinuity, even if infinite in number, have a finite number of so-called limit points.
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