# Download PDF by V. I. Smirnov, A. J. Lohwater: A Course of Higher Mathematics. Integration and Functional

By V. I. Smirnov, A. J. Lohwater

ISBN-10: 1483197476

ISBN-13: 9781483197470

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Sufficiency. Since the points at which a function of bounded variation is continuous are distributed densely in [0,1], we can make use of these points only in forming the Riemann-Stieltjes sum. ^—1) vanish, so that (90) holds for any choice of continuous f(x). The theorem is proved. We have thus shown that the necessary and sufficient condition for integral (89) to give the same linear functional in O as (88) is that the difference h(x) — g(x) be equal to h(0) — g(0) at every point at which it is continuous, as also at x = 1.

There remain the discontinuities at which gr(f — 0) φ g(Ç + 0). If we take

Contained in [a, 6] and consisting of the left-hand end x = a and of all the points x that have rational abscissae. This set of points is dense in [a, 6], and we can extract a subsequence hnjt(x) which is convergent at all the points x^. We thus obtain a limit function h(x), as yet defined only at the points a and x^. e. we put h (x) = sup h (xk). Xk

### A Course of Higher Mathematics. Integration and Functional Analysis by V. I. Smirnov, A. J. Lohwater

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