I need to know the following: let $f:\rightarrow {\mathbb R}$ be a real-analytic function defined in a neighbourhood of a point in an analytic manifold. If the fiber $f^{-1}(0)$ is simply connected, is that so for any fiber $f^{-1}(\epsilon)$ for small $\epsilon$?
In the complex case I gather this is a trivial consequence of Grauert's semicontinuity theorem but what about the real case?