Arc length as a global conformal parameter for analytic curves

We show that arc length is a global conformal parameter for analytic curves and that this parameter can be used to decide whether the domain of definition of an analytic curve can be extended or not. The maximal extension with respect to the arc length parameter is the largest possible extension (over all parametrizations of the curve). Our proof is elementary, simple and short. Several examples are given in the plane, and the results remain true for curves in an arbitrary Euclidean space RkRk.