Abstract
It is reasonably safe to say that any qualified student of mathematics is able to find several texts which adequately treat the subject of convergence of integrals of a sequence of functions over some fixed domain. However, even with the most astute researcher, it is almost impossible to find an adequate treatment of integrals of a function or sequence of functions over a non-fixed domain, e.g. , a sequence of piecewise smooth arcs. If the search is restricted to treatments of discontinuous functions, then the task is even more difficult. This thesis, like Novotny’s, helps to fill this often overlooked gap in the analysis field. | Novotny’s thesis is a treatment of integrals of continuous functions, with respect to arc length, defined on a sequence of arcs which converges to a limiting arc. He establishes sufficient conditions under which those integrals will converge to the corresponding integral on the limiting piecewise smooth differentiable arc or rectifiable arc. He also considered how various alterations of conditions on the sequence of arcs affected convergence of the integrals. This thesis establishes first, properties of the integrals of piecewise smooth differentiable arcs, as well as one important property of a bounded function; and second, sufficient conditions for the convergence of the integrals of a class of functions more general than those considered by Novotny, namely, bounded functions.