Oh, it's actually the case that the roots change smoothly as you smoothly change coefficients, as long as you avoid hitting double roots -- this is an exercise

Oh, it's actually the case that the roots change smoothly as you smoothly change coefficients, as long as you avoid hitting double roots -- this is an exercise in using the implicit function theorem. It's just as you say: the "delta" from the definition of continuity may be much, much smaller in general then the "epsilon". Another thing that might be going on here is that while the roots vary continuously as function of coefficients, the continuity is probably not uniform. There's a whole notion of uniform continuity, which basically says in this case that, regardless of the starting point, if you change coefficients by some delta, the roots won't move further then some global constant time delta. In general, depending on the starting point, when you change the coefficients by delta, the roots may move by 10delta or 100000delta, but uniform continuity would say that this is not the case.