Exercise · ZW-UNIFORM-CONTINUITY-003 · ★☆☆ Basic
Which of the following statements is true?
Choose one option.
Show answer
- A. Every continuous function on \([0,1]\) is uniformly continuous on \([0,1]\).
Explanation
This is the Heine-Cantor theorem: continuous on a closed bounded interval implies uniformly continuous. It fails on \((0,1)\) with \(1/x\) and on \(\mathbb R\) with \(x^2\), and \(f(x)=x\) is both uniformly continuous and differentiable.
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
- ZW-UNIFORM-CONTINUITY-001 For f(x)=2x on mathbb R, which δ (in terms of ε) guarantees |f(x)-f(y)|<ε whenever |x-y|<δ?
- ZW-UNIFORM-CONTINUITY-002 Which of these functions is continuous on (0,1) but not uniformly continuous on (0,1)?
- ZW-UNIFORM-CONTINUITY-004 If f is uniformly continuous on A and (x_n) is a Cauchy sequence in A, what follows?
- ZW-UNIFORM-CONTINUITY-005 If |f(x)-f(y)|≤ L|x-y| for all x,y with L>0, which δ works for every ε>0?
- ZW-UNIFORM-CONTINUITY-006 For f(x)=x^2 on [0,3] we have |f(x)-f(y)|=|x+y| |x-y|. What is the smallest constant L with |f(x)-f…
- ZW-UNIFORM-CONTINUITY-007 To prove sin x is uniformly continuous on mathbb R one uses |sin x-sin y|≤|x-y|. Which δ works?
- ZW-UNIFORM-CONTINUITY-008 Let f(x)=x^2, x_n=n, y_n=n+tfrac1n. What is f(y_n)-f(x_n)?
- ZW-UNIFORM-CONTINUITY-009 Given that |√ x-√ y|≤√|x-y| for x,y≥0, which δ proves √ x is uniformly continuous on [0,∞)?