TITLE: Extension of uniformly continuous function
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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...