  Consider a sequence of cadlag processes $\{X^n\}_n$, and some fixed function
$f$. If $f$ is continuous then under several modes of convergence $X^n\to X$
implies corresponding convergence of $f(X^n)\to f(X)$, due to continuous
mapping. We study conditions (on $f$, $\{X^n\}_n$ and $X$) under which
convergence of $X^n\to X$ implies $\left[f(X^n)-f(X)\right]\to 0$. While
interesting in its own right, this also directly relates (through integration
by parts and the Kunita-Watanabe inequality) to convergence of integrators in
the sense $\int_0^t Y_{s-}df(X^n_s)\to\int_0^t Y_{s-}df(X_s)$. We use two
different types of quadratic variations, weak sense and strong sense which our
two main results deal with. For weak sense quadratic variations we show
stability when $f\in C^1$, $\{X^n\}_n,X$ are Dirichlet processes defined as in
\cite{NonCont} $X^n\xrightarrow{a.s.}X$, $[X^n-X]\xrightarrow{a.s.}0$ and
$\{(X^n)^*_t\}_n$ is bounded in probability. For strong sense quadratic
variations we are able to relax the conditions on $f$ to being the primitive
function of a cadlag function but with the additional assumption on $X$, that
the continuous and discontinuous parts of $X$ are independent stochastic
processes (this assumption is not imposed on $\{X^n\}_n$ however), and
$\{X^n\}_n,X$ are Dirichlet processes with quadratic variations along any
stopping time refining sequence. To prove the result regarding strong sense
quadratic variation we prove a new It\^o decomposition for this setting.
