I hope this isn't considered bumping but . . .
I've been thinking about the problem, and the best idea I could come up with is that the set has something to do with the function f(x)=ln(x). I know for some open sets V in U f^-1(V) is C-open, but I don't think they all are. IS the set found by somehow manipulating this function so that f^-1(V) is C-open for all U-open sets V? Or am I completely off base here?
EDIT: OK, I was thinking about it more, and this seems way too simple, but would f(x)=c where c is a constant be right? This would make the function a line parallel or equal to the x-axis thus the inverse of any U-open set would equal (neg. infinity, pos. infinity) which is C-open or the empty set, which is also C-open correct? But is this ALL possible functions that are C-U continuous?