$\newcommand\P{\mathbb{P}}$Suppose that $g : M \to \mathbb{R}$ is continuous at a single point $x_0\in M$, and the sequence of random variables $X_n \to x_0$ either almost surely or in probability. Can we say that $g(X_n)\to g(x_0)$ in the (a) almost sure or (b) in probability sense?
The standard continuous mapping theorem and proofs thereof assume that $X_n\not\in DC(g)$ almost surely. For the canonical singularly-continuous function $g(x) = x\,\mathbf{1}[x\in\mathbb{Q}]$, (a) holds clearly and (b) holds since $$\begin{align*} \P(|g(X_n)| \geq \epsilon) &\leq \P(|X_n| \geq \epsilon)\to 0 \end{align*}$$ by the Lipschitz property of $g$ at $x=0$.