Removing Isolated Zeroes by Homotopy

Suppose that the inverse image of the zero vector by a continuous map f : R → R has an isolated point P . The existence of a continuous map g which approximates f but is nonvanishing near P is equivalent to a topological property we call “locally inessential,” generalizing the notion of index zero for vector fields, the q = n case. For dimensions n, q where πn−1(Sq−1) is trivial, every isolated zero is locally inessential. We consider the problem of constructing such an approximation g, and show that there exists a continuous homotopy from f to g through locally nonvanishing maps. If f is a semialgebraic map, then there exists such a homotopy which is also semialgebraic. For q = 2 and f real analytic with a locally inessential zero, there exists a Hölder continuous homotopy F (x, t) which, for (x, t) = (P, 0), is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map f , is stated as an open question.