Exercise 6.2.14

Question

A sequence of functions $\left(f_{n}\right)$ defined on a set $E \subseteq \mathbf{R}$ is called equicontinuous if for every $\epsilon>0$ there exists a $\delta>0$ such that $\left|f_{n}(x)-f_{n}(y)\right|<\epsilon$ for all $n \in \mathbf{N}$ and $|x-y|<\delta$ in $E$. \begin{enumerate}[label=(\alph*)] \item What is the difference between saying that a sequence of functions $\left(f_{n}\right)$ is equicontinuous and just asserting that each $f_{n}$ in the sequence is individually uniformly continuous? \item Give a qualitative explanation for why the sequence $g_{n}(x)=x^{n}$ is not equicontinuous on $[0,1]$. Is each $g_{n}$ uniformly continuous on $[0,1]$ ? \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item For equicontinuous functions the same $\delta$ works for every function in the sequence, as opposed to individually being uniformly continuous where $\delta$ depends on $n$.
  \item Not equicontinuous since as $n$ increases we need $\delta$ to be smaller, hence $\delta$ cannot be written independent of $n$. Each $g_n$ is uniformly continuous however (since $g_n$ is continuous on the compact set $[0,1]$).
   \end{enumerate}

Exercise 6.2.14

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Exercise 6.2.14

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