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Is a uniformly continuous function vanishing at $0$ bounded by $a|x|+c$?

On the wikipedia page for Modulus of Continuity, it states that for $f$ uniformly continuous on $\mathbb{R}$, $\exists a,b>0 \in \mathbb{R}$ such that $|f(x)| \leq a|x|+b$ for all $x \in \mathbb{R}$. Can someone please explain why this is?

Regards, MM.

Community
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Mathmo
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1 Answers1

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$f$ is uniformly continuous, thus given $\epsilon>0$ there exists $\delta>0$ such that $|x-y|\leq \delta$ yields $|f(x)-f(y)|\leq \epsilon$. By chosing any sequence $(x_k)_{k=0}^N$ such that $x_0=0$, $x_N=x$ and $|x_{k+1}-x_k|\leq \delta$, we have
$$ |f(x)-f(0)|\leq \sum_{k=0}^{N-1} |f(x_{k+1})-f(x_k)|\leq N\epsilon. $$ But since $|x|/\delta\geq N$, you indeed obtain

$$ |f(x)|\leq (\epsilon/\delta)|x|+|f(0)|.$$

Student
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  • Oops, I've seen the Jonas Meyer's comment after posting. – Student Jan 29 '12 at 21:37
  • There is a mistake. It is not always possible to take $b=|f(0)|$. For example, consider $f(x)=\sqrt[3]{x}$, which is uniformly continuous, but for which $|f(x)|\leq a|x|$ is impossible. I suggest revisiting the assumption that $|x|/\delta\geq N$. – Jonas Meyer Jan 29 '12 at 22:12