Unstability problem of real analytic maps

As well-known, the $C^{\mathrm{\infty }}$ stability of proper $C^{\mathrm{\infty }}$ maps is characterized by the infinitesimal $C^{\mathrm{\infty }}$ stability. In the present paper we study the counterpart in real analytic context. In particular, we show that the infinitesimal $C^{\omega }$ stability does not imply $C^{\omega }$ stability; for instance, a Whitney umbrella ${\mathbb{R}}^2\to {\mathbb{R}}^3$ is not $C^{\omega }$ stable. A main tool for the proof is a relative version of Whitney’s Analytic Approximation Theorem which is shown by using H. Cartan’s Theorems A and B.