Above mentioned fact is proved, now does the converse (i.e If $F(x)$ is uniformly continuous on $\mathbb R$ then it is Continuous on $\mathbb R$ and $\lim_{x \to \infty} F(x)$ exists) of the above question holds. if not, please provide a counterexample to hold the ground and if it's converse holds please provide suggestion how can I proceed to prove that, Thank you
Lastly, If a Function is uniformly continuous on an open interval say $(a,b),$ is it necessary that limit on endpoints exist.
Uniform continuity just means that $|f(x) - f(y)|$ is controlled by $|x-y|$ uniformly. However, it is not immediately clear to what degree this controls the growth of $f$.
For example, the function $f(x) = x$ is clearly unbounded and having no limit at infinity, but is uniformly continuous.
One can show, as a corollary of uniform continuity, the following lemma , which reflects the growth of a uniformly continuous function.
Let $F : (a , \infty) \to \mathbb R$ be uniformly continuous. There exists $A ,B > 0$ such that $|F(x)| \leq A|x| +B$ for all $x$. Thus, $F$ has at most linear growth.
So , for example, something like $x^2$ can't be uniformly continuous, since it doesn't have linear growth.
Note : If you include the limit being infinity, as having a limit, then the function $\sin x$ is also uniformly continuous but has no limit at all.
