Exercise · ZW-UNIFORM-CONTINUITY-005 · ★☆☆ Basic
If \(|f(x)-f(y)|\le L|x-y|\) for all \(x,y\) with \(L>0\), which \(\delta\) works for every \(\varepsilon>0\)?
Choose one option.
Show answer
- C. \(\delta=\varepsilon/L\)
Explanation
With \(|x-y|<\varepsilon/L\) we get \(|f(x)-f(y)|\le L|x-y|<\varepsilon\). Every Lipschitz function is uniformly continuous.
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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