THE LOJASIEWICZ EXPONENT OF A CONTINUOUS SUBANALYTIC FUNCTION AT AN ISOLATED ZERO

Let f be a continuous subanalytic function defined in a neighborhood of the origin 0 is an element of R-n such that f has an isolated zero at 0. We describe the smallest possible exponents alpha, beta, theta for which we have the following estimates: vertical bar f(x)vertical bar >= c parallel to x parallel to(alpha), m(f)(x) >= c parallel to x parallel to(beta), m(f)(x) >= c vertical bar f(x)vertical bar(theta) for x near zero, where c > 0 and mf (x) is the nonsmooth slope of f at x. We prove that alpha = beta + 1,theta = beta/alpha. In the smooth case, we have mf (x) = parallel to del f(x)parallel to, and we therefore retrieve a result of Gwoidziewicz, which is a counterpart of the result of Teissier in the complex case.