Rigorous proof for the converse of $\lim_{x \to \infty} f(x) \implies f $ is uniformly continuous
Solution 1:
Given a continuous function $f$, by the Weierstrass theorem we know that for every $a<b\in \mathbb{R}, f$ is uniformly continuous $[a,b]$. if.
the function $f(x)=x$ is not uniformly continuous in $[a,\infty)$, exactly because $lim_{x\to \infty}f(x) \to \infty$.
This means that if the limit of a function $f$ as $x$ approaches infinity exists and and is finite, then $f$ is a unif cont function in $[a,\infty)$.