Metric Spaces: Continuous, Unbounded Functions The following question is from Fred H. Croom's book "Principles of Topology"


Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous, unbounded function. Show that there is a number $t_0$ for which $\{f(nt_0):n$ an integer $\}$ is an unbounded set. 


For reference: $D(A)=\sup\{d(x,y):x,y\in A\}$. Alternatively, $D(A) = \sup A-\inf A$
The idea I have in mind is as follows. We can have $A$ = $\{f(nt_0):n$ an integer $\} \subset \mathbb{R}$. We would have to show that the $D(A)$ does not exist. To do this, I would have to show that $\sup A$ and $\inf A$ do not exist. Am I on the right track? How does continuity apply in this case?

I want to thank you for taking the time to read this question. I greatly appreciate any assistance you provide.  
 A: It’s not entirely clear what you mean by ‘$D(A)$ does not exist’, but it appears that you mean that it’s infinite. Showing that there this happens for at least one $t\in\Bbb R$ would indeed suffice, and you could do that if you could find such an $A$ for which either $\sup A=+\infty$ or $\inf A=-\infty$ (which I suspect is what you mean when you speak of them not existing). (You don’t need both $\sup A=+\infty$ and $\inf A=-\infty$.) However, this is more a matter of restating the problem than of actually having a direction in which to try to move.
Here’s an extended hint to get you started in a direction that will work, although the argument is a bit more complicated than I’d hoped. (I may, of course, simply be missing some simpler approach.)
For $t\in\Bbb R$ let $A_t=\{f(nt):n\in\Bbb Z\}$, and assume that each $A_t$ is bounded. For $k\in\Bbb Z^+$ let $B_k=\{t\in\Bbb R:|f(nt)|\le k\text{ for all }n\in\Bbb Z\}$. In other words, $B_k$ is the set of $t\in\Bbb R$ such that $A_t$ is bounded by $k$. By hypothesis $\Bbb R=\bigcup_{k\in\Bbb Z^+}B_k$. 


*

*Apply the Baire category theorem and some facts about continuous functions to show that there are a $k\in\Bbb Z^+$ and $a,b\in\Bbb R$ such that $a<b$, and $[a,b]\subseteq B_k$. 

*For technical reasons it will be convenient to have $0\notin[a,b]$; show that we may assume this.

*Show that $[na,nb]\cup[-nb,-na]\subseteq B_k$ for each $n\in\Bbb Z^+$.

*Show that there is an $m\in\Bbb Z^+$ such that $nb>(n+1)a$ whenever $n\ge m$, and conclude that $$\Bbb R\setminus\bigcup_{n\ge m}\Big((na,nb)\cup(-nb,-na)\Big)$$ is a compact subset of $\Bbb R$.

*Conclude that $f$ must be bounded.
A: 

The idea I have in mind is as follows. We can have $A$ = $\{f(nt_0):n$ an integer $\} \subset \mathbb{R}$. We would have to show that the $D(A)$ does not exist.


The problem with that proposed solution is that it can work only if you can show that for every number $t_0$, $D(A)$ does not exist.  The problem is to show that for some $t_0$, a conclusion holds.  To say that it holds for every $t_0$ is a substantially stronger claim.
