In my opinion, no, as either I do not understand the question or such functions do not exist.

I have two potential interpretations for your 'dense'. It is not quite clear what you mean by 'generic construction'. I am not sure what you mean with 'bounded variation' (I know in principle what BV is supposed to mean). That some things are not clear to me might actually be my fault/ignorance. But, if I make some guesses regarding what could be meant i arrive at somethinng that cannot exist, however this might well be something else than you meat. So, it is simply not clear to me what is meant. Perhaps it is clear to other people; waiting a bit for more feedback here or on math.SE could be a good startegy.

Right now, I think it is better for the question to stay where it is. The main problem with it is that you are using words without defining them, and the words don't seem to be sufficiently standard that people already know what they mean. For example, you never defined "construction method", and you never defined the "generic" condition.

If you want to communicate meaningfully in mathematics, you should reduce ambiguity as much as reasonably possible. Right now, your question is very floppy, and there are many possible meanings one can assign to it. As a result one has to make a big guess to decide on a meaningful answer, and it is socially unacceptable to ask people to make such guesses.

You may be able to leave a small piece vague, and say "I don't know how to make this part precise, but I would appreciate suggestions." If the rest of the question is precise enough to reduce the space of possible meanings to something reasonably small, you might still get a helpful answer.