It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the limit function need not be continuous (take f[subscript n](x) = x[superscript n] on [0, 1], for example). Boas has shown that the pointwise limit function of a sequence of continuous real-valued functions defined on the compact interval [a, b] is, nonetheless, continuous on a "dense" subset of [a, b]. In this paper, the notion of uniform convergence at a point is offered as an alternative to the Boas approach in establishing this and, consequently, other results. The arguments stay within the realm of a first proof course in classical mathematical analysis.