= Solution
Let $g:M'\to\mathbb R$ be bounded and continuous. The composition $h=g\circ f$ is bounded and measurable, and every discontinuity point of $h$ is a discontinuity point of $f$. Thus
$$
\mathbb P(X\in D_h)\leq\mathbb P(X\in D_f)=0.
$$
The <Portmanteau theorem> includes the null-discontinuity criterion: if $X_n\xrightarrow dX$ and a bounded measurable function is continuous at $X$ almost surely, then its expectations converge. Hence
$$
\mathbb E[g(f(X_n))]\longrightarrow\mathbb E[g(f(X))].
$$
Since this holds for every bounded continuous $g$, it proves the <continuous mapping theorem> conclusion $f(X_n)\xrightarrow d f(X)$.
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