Slutsky's Theorem

Question

In Slutsky's Theorem's proof as outlined in the link, we can get the general results that $g(X_n,Y_n)\rightarrow_d g(X,c)$ whenever $g$ is continuous. However, in the Continuous Mapping Theorem, it said nothing about $g$ having to be continuous. My question is: Is it still true if I replace the condition of continuous with Borel measurable?