In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables.
A continuous function, in Heine's definition, is such a function that maps convergent sequences into convergent sequences: if xn → x then g(xn) → g(x).
The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {xn} with a sequence of random variables {Xn}, and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables.
[2] Meanwhile, Denis Sargan refers to it as the general transformation theorem.
[3] Let {Xn}, X be random elements defined on a metric space S. Suppose a function g: S→S′ (where S′ is another metric space) has the set of discontinuity points Dg such that Pr[X ∈ Dg] = 0.
Then for any δ > 0 consider the set Bδ defined as This is the set of continuity points x of the function g(·) for which it is possible to find, within the δ-neighborhood of x, a point which maps outside the ε-neighborhood of g(x).
By definition of continuity, this set shrinks as δ goes to zero, so that limδ → 0Bδ = ∅.
In terms of probabilities this can be written as On the right-hand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {Xn}.