Exercise · ZW-UNIFORM-CONTINUITY-002 · ★☆☆ Basic
Which of these functions is continuous on \((0,1)\) but not uniformly continuous on \((0,1)\)?
Choose one option.
Show answer
- C. \(1/x\)
Explanation
The other three extend continuously to \([0,1]\), hence are uniformly continuous on \((0,1)\). But \(1/x\) blows up near \(0\): no single \(\delta\) works for all points at once.
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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