We construct a variety of mappings from the unit interval I into Lp([0,1]); 1 ≤ p > ∞ to generalize classical examples of Lp-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 Lp-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.
|Number of pages||16|
|Journal||Real Analysis Exchange|
|Publication status||Published - 2013|