Distribution convergence proof for strictly monotone map

How can I prove that $X_n \rightarrow _d X$, where $X_n \in \mathbb{R}$ and $X \in \mathbb{R}$ for any strictly monotone and potentially discontinous map $f: \mathbb{R} \rightarrow \mathbb{R}$, we have $f(X_n)\rightarrow _d f(x)$?

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Seems difficult to prove: try $X_n=1/n$ with full probability and $f:x\mapsto x+\mathbf 1_{x\gt0}$.