Abstract
Abstract As far as the development of integration theory is concerned much of this chapter is a side-track. It rounds off the study of continuous functions on compact intervals, investigating limits and infinite sums of such functions, and in particular power series. The arguments are quite elementary, but firmly rooted in the ε-δtradition. Our main technique is that of uniform convergence. We stress that uniform convergence rarely occurs (except locally) for sequences of functions on unbounded intervals. This makes it of limited value in our general theory. The powerful and widely applicable convergence theorems proved in Chapter 14 subsume the elementary results on limits of integrals we derive in this chapter.
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