[math-fun] backwards calculus-type problem

We all know that f'(x)=df(x)/dx is the limit of [f(x+deltax)-f(x)]/deltax as deltax approaches zero. However, what if I'm interested in a situation where deltax>0 doesn't approach zero. Suppose I have a variable y=f(x) and I want to consider a _fixed_ deltay. Is there any straightforward way to find an x such that deltax is minimized, yet deltay = f(x+deltax)-f(x) ??? There may be no such pair (x,deltax), or there may be a multiplicity of x's with the same minimal deltax. I'm primarily interested in finding one, but finding them all would be even better.