Exercise · ZW-UNIFORM-CONTINUITY-006 · ★★☆ Intermediate
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 \(|f(x)-f(y)|\le L|x-y|\) on \([0,3]\)?
Choose one option.
Show answer
- B. \(L=6\)
Explanation
On \([0,3]\) the factor \(|x+y|\) is at most \(6\), so \(L=6\) works and is the smallest possible (approached with \(x,y\) near \(3\)). Hence \(\delta=\varepsilon/6\) works. \(L=3\) and \(L=5\) fail; \(L=9\) works but is not the smallest.
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-003 Which of the following statements is true?
- 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-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,∞)?