A pointwise convergence theorem for sequences of continuous functions.

Abstract:

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