Skip to main content

Calculus Small Changes, Big Ideas

Bill Shillito

Section 25.3 Pointwise and uniform convergence

Introduction goes here.
🔗
  • If, for each \(k\in\bb N\text{,}\) we have a function \(f_{k}:A\to\bb R\text{,}\) then
    \begin{equation*} f_{1},f_{2},f_{3},f_{4},\cdots \end{equation*}
    is a sequence of functions and is denoted \((f_{k})\text{.}\)
    🔗
    🔗
  • Suppose \((f_{k})\) is a sequence of functions, each defined on \(A\subseteq\bb R\text{.}\) The sequence \((f_{k})\) of functions converges pointwise to a function \(f:A\to\bb R\) if, for each \(a\in A\text{,}\)
    \begin{equation*} \limit{k\to\infty}f_{k}(a)=f(a)\text{.} \end{equation*}
    🔗
    🔗
  • We say \((f_{k})\) converges uniformly to \(A\) to a function \(f\) if, for every \(\eps>0\text{,}\) there exists an \(N\in\bb N\) such that \(|f_{k}(x)-f(x)|<\eps\) for all \(k\ge N\) and for all \(x\in A\text{.}\)
    🔗
    🔗
  • If each \(f_{k}\) is continuous at some \(c\text{,}\) and \((f_{k})\) converges uniformly to \(f\text{,}\) then \(f\) is continuous at \(c\text{.}\)
    🔗
    🔗
🔗
PrevTopNext