Various Named Integrals

As n approaches infinity, \(\mathscr {D}_n\) converges to \(\frac{1}{2}[f(x+0) + f(x-0)]\) , provided that x is in the interior of an interval \((\alpha ,\beta )\) where f(x) is of bounded variation. This convergence is uniform if x is in (a, b) a subinterval entirely interior to \((a_1, b_1)\) , where f(x) is continuous and of bounded variation (see Sects. 7.1 , 7.4 and 7.11 ).