Functions and sequence of random variables

Question

Let $\{X_{n}\}_{n \geq 1}$ be a sequence of random variables. Find a bounded function $g$ that doesn't preserve convergence in probability.

$X_{n} \xrightarrow[]{P} X$ however $g(X_{n}) \not\xrightarrow[]{P} g(X)$

I know by the Continuous Mapping Theorem that functions continuous at all $ X(\omega): \omega \in \Omega$ aren't an option. However, I'm also failing when proposing discontinuous functions. I tried some piecewise functions that take constant values $a$ and $b$ accordingly to the value that $X_{n}$ takes. I don't have measure-theoretic background. Thanks for the help in advance.

Answer 1

Let $g(x)=I_{[0,\infty)}$, $X_n=-\frac 1 n$, $X=0$. Then $X_n \to X$ in probability but $g(X_n)=0$ and $g(X)=1$.