  We construct a variety of mappings of the unit interval into
$\mathcal{L}^p([0,1])$ to generalize classical examples of
$\mathcal{L}^p$-convergence of sequences of functions with simultaneous
pointwise divergence. By establishing relations between the regularity of the
functions in the image of the mappings and the topology of $[0,1]$, we obtain
examples which are $\mathcal{L}^p$-continuous but exhibit discontinuity in a
pointwise sense to different degrees. We conclude by proving an Egorov-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 $[0,1]$ and
establish pointwise limits for all functions in the remaining image.
