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?