My difficulty is the title problem. In the problem that is asked, I am attempting to show that there exists a positive continuous function $f$ on $\mathbb{R}$ so that $f$ is Lebesgue integrable on $\mathbb{R}$, but yet $\limsup_{x\rightarrow\infty} f(x) = \infty$.
The hint (title) tells me how I should construct my function.
I'm stuck because I'm not sure how to make $f$ continuous. I calculated my segment for values of $n=1,2,3$ and know that:
$f(1) = 1$ on $[1,2)$, $f(2) = 2$ on $[2, 17/8)$, and $f(3) = 3$ on $[3,82/27)$.
It's clear that the function's gaps get larger and larger as $n$ grows, and I'm not sure how to remedy this to make $f$ continuous. Hints/ideas would be greatly appreciated!
Thanks, Dom