Arc-analytic roots of analytic functions are Lipschitz

Let $g$ be an arc-analytic function (i.e., analytic on every analytic arc) and assume that for some integer $r$ the function $g^r$ is real analytic. We prove that $g$ is locally Lipschitz; even $C^1$if $r$ is less than the multiplicity of $g^r$. We show that the result fails if $g^r$ is only a $C^k$, arc-analytic function (even blow-analytic), $k\in {\mathbb N}$. We also give an example of a non-Lipschitz arc-analytic solution of a polynomial equation $P(x,y)= y^d +\sum_{i=1}^{d}a_i(x)y^{d-i}$, where $a_i$ are real analytic functions.