Lebesgue Dominated Convergence theorem where $x \to x_0$ instead of $ n \to \infty$?

Question

I had learned the usual dominated convergence theorem which says that if $f_n \to f$, and each $f_n$ bounded by some integrable g, then $\int f_n \to \int f$. However, I came across a "different" version in one of the examples, where dominated convergence was used to justify that if x tends to $x_o$ implies f(x) tends to $f(x_0)$ then $\int f(x) \to \int f(x_0)$. Could someone show why (presumably) the original one implies this one? Thanks!