Analytic-function sequence limits are integrals {Stieltjes integral, Riemann}. Stieltjes integrals generalize simpler integrals {Riemann-Darboux integral}.
Integrals, from x = a to x = b, of f(x) * dg(x) * dx equal limits of sums, from i = 0 to i = n, of f(e(i)) * (g(x(i + 1)) - g(x(i))), where x(i) are partition intervals and e(i) are inside intervals (x(i), x(i + 1)).
Riemann
Functions {Riemann integrable function} can have no discontinuities or have discontinuities that form measure-zero sets. Riemann integrals are Lebesgue integrable, but Lebesgue integrals can be not Riemann integrable.
Mathematical Sciences>Calculus>Analysis>Integral
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Date Modified: 2022.0224