Suppose $f(x)$ is continuous on $\mathbb{R}$, for $\forall \delta>0, \forall x\in\mathbb{R}, \lim\_{n\rightarrow\infty}f(x+n\delta)=+\infty$. Is it correct that $\lim\_{x\rightarrow+\infty}f(x)=+\infty$?



Continuous functions and infinity
Yes, this is true and well known. One of the references I know is the problem book of B. Makarov, M. Goluzina, A. Lodkin and A. Podkorytov (Selected problems in real analysis, Translations of Mathematical Monographs 107, AMS 1992), problem II.1.25 (it is slightly different, but essentially the same).


An idea is that if, on the contrary, there are intervals $\Delta\_k=[x\_k,y\_k]$ onto which $f$ is bounded, $x\_k\to \infty$, then we may construct a nested family of closed intervals $T\_i$, $T\_1\supset T\_2\supset T\_3\dots$, and positive integers $n\_1<n\_2<\dots$, such that each interval of the form $n\_iT\_i$ is contained in some $\Delta\_k$. Then for a common point $t=\cap T\_i$ it appears that $f(nt)$ does not tend to infinity. 


