Exercise · ZW-UNIFORM-CONTINUITY-010 · ★★★ Advanced
For \(f(x)=1/x\) on \((0,1)\), let \(x_n=\tfrac1n\) and \(y_n=\tfrac1{n+1}\). What are \(|x_n-y_n|\) and \(|f(x_n)-f(y_n)|\)?
Choose one option.
Show answer
- A. \(\tfrac1{n(n+1)}\) and \(1\)
Explanation
\(\tfrac1n-\tfrac1{n+1}=\tfrac1{n(n+1)}\to0\), but \(|n-(n+1)|=1\) for all \(n\). The distance between images does not shrink although that between points does: \(1/x\) is not uniformly continuous on \((0,1)\).
Review the concept: Uniform continuity
A function \(f\) is uniformly continuous on a set \(A\) if for each \(\varepsilon>0\) there is a \(\delta>0\), valid for all points at once, that controls \(|f(x)-f(y)|\) by \(|x-y|\) in \(A\).
\[\forall\varepsilon>0\ \exists\delta>0:\ x,y\in A,\ |x-y|<\delta\Rightarrow|f(x)-f(y)|<\varepsilon\]
More practice problems: uniform continuity
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