  Motivated by questions arising in financial mathematics, Dupire introduced a
notion of smoothness for functionals of paths (different from the usual
Fr\'echet--Gat\'eaux derivatives) and arrived at a generalization of It\=o's
formula applicable to functionals which have a pathwise continuous dependence
on the trajectories of the underlying process. We study nonlinear functionals
which do not have such pathwise continuity and further work simultaneously
under the family of continuous semimartingale measures on path-space. We do
this without introducing a second component, as carried out by Cont--Fournie
but by using old work of Bichteler which allows to keep a pathwise picture even
for complex functionals
