Local extrema of analytic functions

We give a complete answer to the problem of the finite decidability of the local extremality character of a real analytic function at a given point, a problem that found partial answers in some works by Severi and {\L}ojasiewicz. Consider a real analytic function $f$ defined in a neighbourhood of a point $x_0\in\Rn$. Restrict~$f$ to the spherical surface centered in~$x_0$ and with radius~$r\ge0$ and take its infimum~$m(r)$ and its supremum~$M(r)$. We establish some properties of~$m(r)$ and~$M(r)$ for small~$r>0$. In particular, we prove that they have asymptotic expansions of the form $f(x_0)+c\cdot(r^\alpha +o(r^\alpha))$ as~$r\to0$ for a real~$c$ and a rational~$\alpha\ge1$ (of course the parameters will usually be different for~ $m$ and~$M$).