Nonvanishing on a neighborhood of a compact set

Statement

Let $f$ be continuous near a compact set $K$ and suppose $|f(x)|\ge c>0$ on $K$ . Then there is an open neighborhood $V$ of $K$ on which $|f|>c/2$ . Continuity supplies a suitable neighborhood at each point; finitely many cover $K$ .

Holomorphic denominators and parameters

For holomorphic $f$ , this gives a region where $1/f$ is holomorphic and bounded. A jointly continuous compact parameter family with the same margin admits a common neighborhood by applying the argument on the compact product. For a noncompact family, uniformity needs a separate estimate. Being nonzero at each real point for each parameter does not by itself furnish a common complex neighborhood or a bounded inverse.