  We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is
a semimartingale. The function $f$ is not required to be differentiable, so
It\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of
$t$, it is shown that requiring $f$ to be locally Lipschitz continuous in $x$
is enough for an It\^{o}-style decomposition to exist. In particular, $Y$ will
be a Dirichlet process. We also look at the case where $f(t,x)$ can depend on
$t$, possibly discontinuously. It is shown, under some additional mild
constraints on $f$, that the same decomposition still holds. Both these results
follow as special cases of a more general decomposition which we prove, and
which applies to nondifferentiable functions of Dirichlet processes. Possible
applications of these results to the theory of one-dimensional diffusions are
briefly discussed.
