The operator of inversion as an everywhere continuous nowhere differentiable function

The present paper is devoted to investigating the inversion operator between various spaces of continuously differentiable functions. Up to a certain critical order — existence and formulae for its higher order derivatives are presented. Beyond this critical order — the inversion operator turns out to be nowhere differentiable. In particular, operator f → f−1 as a self-homeomorphism of Diffr(M), the Banach manifold of Cr(r = 1, 2, …) self-diffeomorphisms of a compact manifold M, is nowhere Fréchet differentiable. A local characterization of higher order Gateaux differentiability of operator f → f−1 is also given. © 2000, Birkhäuser Verlag, Basel.