Relations between L^p-and pointwise convergence of families of functions indexed by the unit interval

We construct a variety of mappings from the unit interval I into L^p([0,1]); 1 ≤ p > ^∞ to generalize classical examples of L^p-converging equences of functions with simultaneous pointwise divergence. By es-tablishing relations between the regularity of the functions in the image of the mappings and the topology of I, we obtain examples which are L^p-continuous but exhibit discontinuity in a pointwise sense to different egrees. We conclude by proving a Lusin-type theorem, namely that if almost every function in the image is continuous, then we can remove a set of arbitrarily small measure from the index set I and establish pointwise continuity in the remainder.