Objectives
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Extend uniformly continuous functions to closures of their domains.
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Compare continuity classes using Lipschitz conditions and related implications.
Section 24.2 Uniform continuity
Introduction goes here.
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We say \(f:A\to\bb R\) is uniformly continuous if for all \(\eps>0\) there exists some \(\delta>0\) such that for all \(x,y\in A\text{,}\) \(|x-y|<\delta\) implies \(|f(x)-f(y)|<\eps\text{.}\) Notable, the same \(\delta\) works everywhere.
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A continuous function on a compact set is uniformly continuous.
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A uniformly continuous function has a uniformly continuous extension to the closure of that set. This helps eventually show that \(2^{\pii}\) exists, by extending \(2^{x}\) from a function on \(\bb Q\) to a function on \(\bb R\text{.}\)
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A function is Lipschitz continuous if there is a positive real constant \(K\) such that, for all real \(x_{1}\) and \(x_{2}\text{,}\)\begin{equation*} \abs{f(x_1)-f(x_2)}\le K\abs{x_1-x_2} \end{equation*}or put another way\begin{equation*} \abs{\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}}}\le K\text{.} \end{equation*}The value of \(K\) is called the Lipschitz constant or the modulus of (uniform) continuity. Lipschitz functions are “almost differentiable.”
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Continuously differentiable \(\to\) Lipschitz continuous \(\to\) uniformly continuous \(\to\) continuous.
