Let E,F,G be 3 banach spaces, U an open set of E and f:U->F , g:U->G be 2 continously differentiable applications.

Suppose that $\displaystyle

\forall x \in U\,Df(x)$ is bijective. Show that f(U) is an open set in F

then suppose that $\displaystyle

\exists \varphi :f(U) \to G\,such\,that\,g = \varphi \circ f

$

show that $\displaystyle \varphi $ is continously differentiable