Abstract

It is shown to be consistent that the reals are covered by $\aleph_1$ meagre sets yet there is a Baire class 1 function which cannot be covered by fewer than $\aleph_2$ continuous functions. A new cardinal invariant is introduced which corresponds to the least number of continuous functions required to cover a given function. This is characterized combinatorially. A forcing notion similar to, but not equivalent to, superperfect forcing is introduced.