What is a "banded function"?

@researcher: what leads you to suspect such control can be obtained if $k(x) \geq 0$?

Looks like researcher edited his question. Cool!

I meant "progress", not "orogress" !

This is an interesting question but the $z$-$y$ ambiguity needs resolution to allow orogress. Waiting for the OP to speak up . . .

@Dylan Wilson: +1 on your comment which, as far as I know, is the first occurrance of the term "love magic" in a serious mathematical context. And now I beg you to excuse me, I have to get back to some differential equation love magic!