# If $f(x)$ is uniformly continuous at $(0,1)$ then is it bounded at $(0,1)$? [duplicate]

Possible Duplicate:
Uniform continuity

I have a question which is:

If ${f(x)}$ is uniformly continuous at ${(0,1)}$ then is it bounded at ${(0,1)}$?

This sound like it's correct to me but I can't see why exactly (Or maybe it's wrong :P).

Could someone help me figure out the truth here? :) Thanks!

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Hint: fix $\epsilon>0$ and make $n\delta$ steps from $\frac12$ until you reach any bound of $(0,1)$. –  Ilya Dec 2 '11 at 16:25
"on" not "at"... –  David Mitra Dec 2 '11 at 16:27

## marked as duplicate by Jonas Meyer, t.b., Asaf Karagila, Zhen Lin, Ｊ. Ｍ.Dec 16 '11 at 2:02

By uniform continuity there is an $n\in{\mathbb N}_{\geq 1}$ such that $0<x\leq y<x+{1\over n}<1$ implies $|f(y)-f(x)|<1$. Put $x_k:={k\over n+1}$ $\ (1\leq k\leq n)$. Then any $y \in \ ]0,1[\$ is at distance ${}<{1\over n}$ from an $x_k$. Therefore we have

$$|f(y)| < \max_{1\leq k\leq n} |f(x_k)| + 1\ =:\ M$$

for all $y \in \ ]0,1[\$.

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$f$ would indeed be bounded.
$f$ is continuous on $(0,1)$; thus, if it were not bounded on $(0,1)$, there would exist $x_n\nearrow 1$ or $x_n\searrow 0$ with $f(x_n)\rightarrow\infty$ or $f(x_n)\rightarrow-\infty$. But, none of these options can happen due to the following facts: 1) uniformly continuous functions map Cauchy sequences to Cauchy sequences. 2) Cauchy sequences are bounded.
Fact 1) is easily proven: Let $\{x_n\}$ be Cauchy and $\epsilon>0$. Let $\delta$ be such that $|f(x)-f(y)|\lt\epsilon$ whenever $|x-y|<\delta$. Now choose $N$ so that $m,n> N$ implies $|x_n-x_m|<\delta$. We then have for any $n, m>N$ that $|f(x_n)-f(x_m)|<\epsilon$. Thus, $\{f(x_n)\}$ is Cauchy.