the continuous mapping theorem
If X_n converges in distribution to X, what happens to g(X_n) for a function g? In Volume I you saw this for continuous g on the line. The continuous mapping theorem generalizes it to maps between metric spaces and, crucially, relaxes 'continuous everywhere' to 'continuous where it matters'. It is the tool that lets you push weak convergence through transformations, the engine behind reading off limit laws of functionals (maxima, integrals, hitting times) from a functional limit theorem.
Let S, S' be metric spaces and g: S -> S' a measurable map. Let D_g be the set of discontinuity points of g (a Borel set). The theorem states: if X_n => X in S and the limit law assigns no mass to the discontinuity set, P(X in D_g) = 0, then g(X_n) => g(X) in S'. In words, the image laws converge as long as the limit almost surely lands where g is continuous. The proof is one line via Skorokhod representation: realize Y_n -> Y a.s. with the right laws; since Y avoids D_g a.s., continuity gives g(Y_n) -> g(Y) a.s., hence g(X_n) => g(X). Equivalently one checks E[f(g(X_n))] -> E[f(g(X))] for f in C_b(S') using that f composed with g is continuous off D_g.
The relaxation to a.e.-continuity is what makes the theorem so useful in path space. The supremum functional X |-> sup_t X(t) on C[0,1] is continuous, but on D[0,1] (the cadlag path space with the Skorokhod topology) functionals like 'value at a fixed time t' are discontinuous exactly at paths jumping at t; the theorem still applies because a Brownian or Levy limit jumps at a fixed deterministic time with probability zero. This is how Donsker's theorem instantly yields the limit law of the maximum of a random walk: apply the continuous sup functional to the Brownian limit.
If sqrt(n)(X_bar_n - mu) => N(0, sigma^2) (the CLT), apply the continuous map g(x) = x^2: then n(X_bar_n - mu)^2 => sigma^2 * chi^2_1, since g is continuous everywhere so D_g is empty. This is the delta-method's simplest instance, and it is just the continuous mapping theorem applied to a smooth function of a Gaussian limit.
Pushing a CLT through a continuous map to get the limit law of a transformed statistic.
The hypothesis P(X in D_g) = 0 cannot be dropped. If the limit puts positive mass on a discontinuity of g, the conclusion can fail outright; mere measurability of g is not enough.