the continuous mapping theorem

Once you have established that some random sequence converges, a natural next question is: what about a transformed version of it? If X_n converges to X, does g(X_n) converge to g(X) for a function g of interest — its square, its log, its reciprocal? The continuous mapping theorem gives a clean, permissive yes: continuous functions carry convergence along with them, in all the usual modes.

Precisely: if g is a continuous function (or continuous at least on the set where X actually lands, with probability 1), then convergence of X_n to X is preserved under g. If X_n converges in distribution to X, then g(X_n) converges in distribution to g(X); the same holds for convergence in probability and almost sure convergence. The intuition is exactly the deterministic fact that continuous functions preserve limits — if x_n approaches x then g(x_n) approaches g(x) — lifted to random variables. You apply your favourite g and the convergence rides through untouched.

This humble-sounding result is a workhorse. It is how you turn a CLT statement about a sum into one about, say, its square (giving the chi-squared limits behind variance tests), or how you justify plugging a consistent estimator into a continuous formula. The one requirement to respect: g must be continuous WHERE it matters. A function with a jump or a blow-up exactly at the limit point can break the conclusion — for instance g(x) = 1/x is fine if X is never 0, but dangerous if the limit can sit on the discontinuity at 0.