Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 3 / 3H / Solution

A real-valued function $f$ on an interval $I$ is uniformly continuous whenThe essential point in uniform continuity is that $\delta$ is independent of the locations of $x$ and $y$. Uniform continuity on the whole real line does not imply boundedness: $f(x)=x$ is Lipschitz continuous with constant $1$, hence uniformly continuous, but is unbounded.