How to show example where convergence in $L^1$ norm does not hold for continuous functions of the random variables?

Question

Suppose a sequence of random variables $X_n$ converges to a random variable $X$ in $L^1$ norm, and that $g: \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function.

It is not necessarily true that $g(X_n)$ converges to $g(X)$ in $L^1$ norm. I'm trying to figure out a counterexample that shows this.

Answer 1

Let $x_n = n 1_{[0,{1 \over n^2}]}$ and $g(t) = t^2$.

Then $\|x_n-0\|_1 \to 0$ but $\|g\circ x_n - g \circ 0 \|_1 = 1$ for all $n$.