We continue our investigations into absolute CR-epic spaces. Given a continuous function f:X→ Y, with X absolute CR-epic, we search for conditions which imply that Y is also absolute CR-epic. We are particularly interested in the cases when X is a dense subset of Y and when f is a quotient mapping. To answer these questions, we consider issues of local extension of continuous functions. The results on this question are of independent interest.

The first and third authors would like to thank NSERC of Canada for its support of this research. We would all like to thank McGill and Concordia Universities for partial support of the middle author's visits to Montreal.