This question comes from the following problem:
A real-valued function $f$ defined on $\mathbb{R}$ has the following property: For every positive $\epsilon$, there exists positive $\delta$ such that, $|f(x) - f(1)|\ge \epsilon$ whenever $|x-1|\ge\delta$.
This property is equivalent to which of the following statements?
(C). $f$ is unbounded
(D). $\lim_{|x|\to\infty} |f(x)|=\infty$
The answer is D and this is obvious if you write down the definition. However, I am trying to understand why C is not correct. Noticing that D implies C, yet C is not the correct answer, it has to be the case that C does not imply D. If my logic is right, does that mean we can find some $f$ defined on $\mathbb{R}$ such that it is unbounded but $\lim_{|x|\to\infty} |f(x)| \ne \infty$?
I cannot think of an example. I thought about $f(x)=\frac{1}{x}$ but it is not defined at $x=0$. Perhaps something is wrong with my logic here?