Exercise · ZW-UNIFORM-CONTINUITY-008 · ★★☆ Intermediate
Let \(f(x)=x^2\), \(x_n=n\), \(y_n=n+\tfrac1n\). What is \(f(y_n)-f(x_n)\)?
Choose one option.
Show answer
- D. \(2+\tfrac1{n^2}\)
Explanation
\((n+\tfrac1n)^2-n^2=2+\tfrac1{n^2}\ge2\), even though \(|y_n-x_n|=\tfrac1n\to0\). So no \(\delta\) works for \(\varepsilon=2\): \(x^2\) is not uniformly continuous on \(\mathbb R\).
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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