Convergence in distribution of a continuous function of random variables

Question

I'm having trouble coming up with a counterexample to the following.

If $X_n$ and $Y_n$ converge in distribution to $X$ and $Y$ respectively (where $X_n$ and $Y_n$ are possibly dependent), and $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is a continuous function, then $f(X_n, Y_n)$ converges in distribution to $f(X, Y)$.

A comment on this question would suggest that one exists, but I'm not sure what sequences or functions would be good places to start thinking.

Answer 1

Let $f$ be a continuous function satisfying $f(1,1)=1$, and $f(0,0)=f(0,1)=f(1,0)=0$.

Let $(X_n,Y_n)=(0,1)$ with probability $1/2$ and equal $(1,0)$ with probability $1/2$, for each $n\ge 1$.

Let $(X,Y)$ be iid distributed over the four corners of $[0,1]^2$.

Note that $X_n\stackrel{d}=X$ and $Y_n\stackrel{d}=Y$. However, $f(X_n,Y_n)=0$ always, while $f(X,Y)=1$ with probability $1/4$.