  The It{\^o} map assigns the solution of a Rough Differential Equation, a
generalization of an Ordinary Differential Equation driven by an irregular
path, when existence and uniqueness hold. By studying how a path is transformed
through the vector field which is integrated, we prove that the It{\^o} map is
H{\"o}lder or Lipschitz continuous with respect to all its parameters. This
result unifies and weaken the hypotheses of the regularity results already
established in the literature.
