Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I want to prove this:

$f\in C((a,b))$ uniformly continuous. Then there exists $\tilde{f}\in C([a,b])$ extension of $f$.

I took $x_n\rightarrow a$ and defined $\tilde{f}(a)=\mathrm{lim}\;f(x_n)$. I saw that this is a good definition, the only thing that I'm not able to prove is that $\tilde{f}$ is continuous at $a$ (or $b$). Could you help me please?

share|cite|improve this question
Note that $f$ continuous is equivalent to $f(lim x_n)=lim f(x_n)$ holds for every convergent sequence. – azarel Jan 19 '12 at 0:42
up vote 4 down vote accepted

You need to justify that $\lim f(x_n)$ exists. It does exist, since uniformly continuous functions map Cauchy sequences to Cauchy sequences (you might need to prove this; but it follows directly from the definitions). Now show that for any sequence $y_n$ converging to $a$, that $ f(y_n)$ converges to $ \tilde f(a)$. This will show that $\tilde f$ is continuous at $a$. Of course, you have to deal with $b$ too...

share|cite|improve this answer

You need to show the limit exists. Do this by verifying that $f(x_n)$ is a Cauchy sequence. Use the uniform continuity to do this.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.