Does a continuous function of real numbers preserve continuity of random variables?

Question

In particular, if $X_1, X_2, \ldots$ is a sequence of random variables such that $X_n\to X$ as $n\to\infty$, does it follow that if $f$ is continuous function (over reals) that $\lim_{n\to\infty} f(X_n) = f(X)$? What if $X$ is constant, is this still true?

Answer 1

Let $\epsilon$ and $\delta$ be positive real numbers. Define the set $$ A_\delta = \{x\in \mathbb{R}\mid \exists y\in \mathbb{R}\ s.t. |x-y|<\delta, |g(x)-g(y)|>\epsilon\} $$

Note that $P(|g(X_n)-g(X)|>\epsilon) \leq P(|X_n-X|\geq\delta)+P(X\in A_\delta)$. So, taking the superior limit as $n\to\infty$, we get $$ \overline{\lim_{n\to\infty}}P(|g(X_n)-g(X)|>\epsilon)\leq P(X\in A_\delta), $$

for every $\delta>0$. Because the continuity of $g$, we have that $\lim_{\delta\to 0^+}P(X\in A_\delta) =0$. And we are done.