Exercise · ZW-UNIFORM-CONTINUITY-004 · ★☆☆ Basic
If \(f\) is uniformly continuous on \(A\) and \((x_n)\) is a Cauchy sequence in \(A\), what follows?
Choose one option.
Show answer
- D. \((f(x_n))\) is Cauchy.
Explanation
Given \(\varepsilon\), the uniform \(\delta\) and an index with \(|x_n-x_m|<\delta\) give \(|f(x_n)-f(x_m)|<\varepsilon\). That \((x_n)\) converges in \(A\) does not follow: the limit may lie outside \(A\).
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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