State the definition of uniform continuity. Give an example of a function \(f:(0,1)\to\mathbb R\) which is continuous but not uniformly continuous.
Solution
1. Definition of uniform continuity.Proof: \(f : D \subseteq \mathbb{R} \to \mathbb{R}\) is uniformly continuous on \(D\) if for every \(\epsilon > 0\) there is \(\delta > 0\) such that for all \(x, y \in D\), \(|x - y| < \delta \implies |f(x) - f(y)| < \epsilon\).(The difference from plain continuity: the same \(\delta\) must work for all pairs \(x, y\), independent of where they sit.)
2. Example: \(f(x) = 1/x\) on \((0,1)\) is continuous but not uniformly continuous.
2.1. Continuous: \(f\) is a quotient of continuous functions with nonzero denominator on \((0,1)\), hence continuous at every point of \((0,1)\).
2.2. Not uniformly continuous: take \(x_n = 1/n\) and \(y_n = 1/(2n)\).Then \(x_n, y_n \in (0,1)\) and \(|x_n - y_n| = 1/(2n) \to 0\), but \(|f(x_n) - f(y_n)| = |n - 2n| = n \to \infty\).Hence no \(\delta > 0\) can control pairs \((x_n, y_n)\) with \(|x_n - y_n| \to 0\), so the uniform-continuity definition fails (with any \(\epsilon\), say \(\epsilon = 1\)).
3. Q.E.D.