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Question: Let $X=X'=\mathbb{R}$ with the usual metric.

(a) Show that a polynomial function $p(x)$ on $\mathbb{R}$ is uniformly continuous if and only if $\deg(p(x))<2$.

So I thought all polynomial functions were continuous? Clearly all polynomial functions regardless of degree preserve limits so that if $x_n\rightarrow x$, $p(x_n)\rightarrow p(x)$, hence continuous?

(b) Show that $f(x)=\sin(x)$ is uniformly continuous on $\mathbb{R}$.

For this one, I have to show that given $\epsilon>0$, exists $\delta>0$ s.t. for any $x,y\in\mathbb{R}$, $d'(f(x),f(y))<\epsilon$ whenever $d(x,y)<\delta$. The important thing is that $\delta$ is independent of the point in $\mathbb{R}$. Not sure how to find this $\delta$ though.

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    It's good to keep in mind that uniform continuity, while equivalent to continuity when your domain is compact, is much stronger than continuity in general. As a warm-up, it's good to prove that $x^2$ is _not_ uniformly continuous on the real line: for a given $\varepsilon$, you have to choose smaller and smaller $\delta$s as you move away from $0$.2012-02-06

3 Answers 3