  We show that non continuous Dirichlet processes, defined as in \cite{NonCont}
are closed under a wide family of locally Lipschitz continuous maps (similar to
the time-homogeneous variants of the maps considered in \cite{Low}) thus
extending Theorem 2.1. from that paper. We provide an It\^o formula for these
transforms and apply it to study of how $[f(X^n)-f(X)]\to 0$ when $X^n\to X$
(in some appropriate sense) for certain Dirichlet processes $\{X^n\}_n$, $X$
and certain locally Lipschitz continuous maps. We also consider how
$[f_n(X^n)-f(X)]\to 0$ for $C^1$ maps $\{f_n\}_n$, $f$ when $f_n'\to f'$
uniformly on compacts. For applications we give examples of jump removal and
stability of integrators.
