Paper 4, Section II, G

What does it mean to say that a function $f$ on an interval in $\mathbf{R}$ is uniformly continuous? Assuming the Bolzano-Weierstrass theorem, show that any continuous function on a finite closed interval is uniformly continuous.

Suppose that $f$ is a continuous function on the real line, and that $f(x)$ tends to finite limits as $x \rightarrow \pm \infty$; show that $f$ is uniformly continuous.

If $f$ is a uniformly continuous function on $\mathbf{R}$, show that $f(x) / x$ is bounded as $x \rightarrow \pm \infty$. If $g$ is a continuous function on $\mathbf{R}$ for which $g(x) / x \rightarrow 0$ as $x \rightarrow \pm \infty$, determine whether $g$ is necessarily uniformly continuous, giving proof or counterexample as appropriate.

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