2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126112For k=1,2,... infty and a Frolicher-Kriegl order k Lipschitz differentiable map f:E supseteq U to E having derivative at x_0 in U a linear homeomorphism E to E and satisfying a Colombeau type tameness condition, we prove that x_0 has a neighborhood V subseteq U with f|V a local order k Lipschitz diffeomorphism. As a corollary we obtain a similar result for Keller C_c^{\infty} maps with E in a class including Frechet and Silva spaces. We also indicate a procedure for verifying the tameness condition for maps of the type x mapsto varphi circ [id,x] and spaces E=C^{\infty}(Q) when Q is compact by considering the case Q=[0,1]. Our considerations are motivated by the wish to try to retain something valuable in an interesting but defective treatment of integrability of Lie algebras by J. Leslie.AmSLaTeX, 8 pages; v2: scope of Corollary 9 extended, misprints corrected; v3: inaccuracies in 3 Def:s, misprints corrected, reorganization of proofs suggested in the former footnote; v4: minor specifications added, misprints corrected; v5: a forgotten detail added in the proof of 10 Prop., minor rewording in the proof of 8 ThmFunctional Analysis58C15, 46A61, 46T20text