  Using Dupire's notion of vertical derivative, we provide a functional
(path-dependent) extension of the It\^o's formula of Gozzi and Russo (2006)
that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is
motivated and illustrated by its applications to the hedging or superhedging
problems of path-dependent options in mathematical finance, in particular in
the case of model uncertainty
