I was reading David Williams' $\textit{Probability with Martingales}$ and in it he mentions that for real-valued random variables, if $X_{n} \to X$ almost surely, then the associated distribution functions $F_{X_{n}}$ converge to $F_{X}$. The proof uses the fact that, since $X_{n} \to X$ almost surely, $f(X_{n}) \to f(X)$ almost surely for all bounded continuous functions. Hence, by the Bounded Convergence theorem, $\int_{\mathbb{R}} f d\mu_{n} \to \int_{\mathbb{R}} f d \mu$, where $\mu_{n},\mu$ are the laws of $X_{n}$ and $X$ respectively.
My question is, doesn't the above proof only indicate that the laws converge weakly? For the distribution functions to converge, shouldn't continuity conditions be considered?
Yes, $X_n \to X$ almost surely implies only $F_{X_n}(t) \to F_X(t)$ for all continuity points $t$ of $F_X$. We cannot expect convergence for the non-continuity points.
Consider, for instance, $$X_n := 1+ \frac{1}{n} \quad \text{and} \quad X=1.$$ Clearly, $X_n \to X$ almost surely. Since $F_{X_n}(1)=0$ for all $n \in \mathbb{N}$ and $F_X(1)=1$, pointwise convergence $F_{X_n}(1) \to F_X(1)$ fails to hold.