  We call a local homeomorphism $f: (R^n,0)\to(R^n,0)$ blow-analytic if it
becomes real analytic after composing with a finite number blowings-up with
smooth nowhere dense centers. If the graph of $f$ is semi-algebraic then, by a
theorem of Bierstone and Milman, $f$ is blow-analytic if and only if it is
arc-analytic: the image by $f$ of a parametrized real analytic arc is again a
real analytic arc.
  For a semialgebraic homeomorphism $f$ we show that if $f$ is blow-analytic
and the inverse of $f$ is Lipschitz, then $f$ is Lipschitz and the inverse of
$f$ is blow-analytic. The proof is by a motivic integration argument, using
additive invariants on the spaces of arcs.
