`  :::Right; I was wrong, I think. So then: Thm: if f(s) has a zero inside of U, then *every* sufficiently nearby function g(s) will also have a zero inside U. This sounds like a perfectly plausible theorem that would seem to date to the classical period, but for the life of me I can't think of a name for it. Some variant on the ``three circles theorem``, or something like that?   Again, its late at night, so I'm babbling.   `