Let $X\_t$ be a fixed cadlag semi-martingale and $J\_n$ be a fixed sequence of functions from $\mathbb{R}^d$ to $\mathbb{R}$ which are twice continuously differentiable. If $J\_n$ converge pointwise to the function $J$ which itself is a.e. differentiable on $\mathbb{R}^d$ (wrt to the Lebesgue measure-zero sets) and *not* differentiable everywhere; then under what non-trivial conditions on $J\_n$ does
$$
J\_n(X\_t) \to J(X\_t),
$$ 
in the Skorohod topology?



Convergence of function of stochastic processes
According the definition of Skorokhod topology, for the $J\_n(X\_t)\to J(X\_t)$ in Skorokhod topology,  

it suffices to suppose that
$$ \lim\_{n\to\infty}\sup\_{\|x\|\le a}|J\_n(x)-J(x)|=0, \qquad \forall a>0.$$ 
i.e. $J\_n $ convergence to $J$ uniformly in every compact set.


