A pointwise convergence theorem for sequences of continuous functions.

Abstract

Let { f k } \{ {f_k}\} be a sequence of continuous real valued functions defined on an interval I I and N N a fixed nonnegative integer such that if f k ( x ) = f i ( x ) {f_k}(x) = {f_i}(x) for more than N N distinct values of x ∈ I x \in I then f k ( x ) ≡ f i ( x ) {f_{k}}(x) \equiv {f_i}(x) for x ∈ I x \in I . It follows that there is a subsequence { g j } \{ {g_j}\} of { f k } \{ {f_k}\} such that for each x x the subsequence { g j ( x ) } \{ {g_j}(x)\} is eventually monotone. Thus lim j → + ∞ g j ( x ) = f ( x ) {\lim _{j \to + \infty }}{g_j}(x) = f(x) exists for all x x , where f f is an extended real valued function. If | f k ( x ) | |{f_k}(x)| is bounded for each x ∈ I x \in I then lim j → + ∞ g j ( x ) = f ( x ) {\lim _{j \to + \infty }}{g_j}(x) = f(x) exists as a finite limit for all x ∈ I x \in I . For N = 0 N = 0 this reduces to picking a monotone subsequence from a sequence of continuous functions whose graphs are pairwise disjoint.