The Continuity Theorem is the bridge between algebraic calculations with moment generating functions and probabilistic conclusions about convergence in distribution. It says that if the MGFs of a sequence of random variables converge pointwise to the MGF of a limit variable (in some neighbourhood of the origin), then the distributions converge: $X_n \xrightarrow{d} X$. In other words, convergence of MGFs implies convergence of CDFs at all continuity points. This is the tool that converts the Central Limit Theorem from a formal MGF calculation into an actual distributional statement. To prove that $(S_n - n\mu)/(\sigma\sqrt{n}) \xrightarrow{d} N(0,1)$, one shows that the MGF of the normalised sum converges to $e^{\theta^2/2}$ — a computation that uses [MGF of a Sum](/theorems/1144) and a Taylor expansion of the individual MGFs — and then the Continuity Theorem delivers the conclusion. Without it, MGF convergence would be an algebraic curiosity with no probabilistic content. The theorem requires that the limit MGF $M_X(\theta)$ be finite in an open interval around $0$, which rules out heavy-tailed limits. For distributions without MGFs (such as the Cauchy distribution), the analogous role is played by characteristic functions and Lévy's continuity theorem, which imposes no moment conditions but requires complex analysis. The [Uniqueness of the MGF](/theorems/1142) is an essential ingredient: without knowing that the limit MGF determines a unique distribution, pointwise convergence of MGFs would be ambiguous.