Consider a continuous variable with support $\mathbb{R}$ and density $f$. For every bounded interval belonging to $\mathbb{R}$, $f$ must be bounded (by continuity). If $f$ is strictly increasing in its unbounded support, then the limit $\lim_{x\rightarrow\infty}f\left(x\right)$ must not exist (otherwise $f$ would not integrate to $1$).
Is there a concrete example of such an $f$, i.e. continuous, atomless and strictly increasing density over an unbounded support? I think I was able to find such a distribution by prior search in a book but I can't find it anymore and would appreciate any directions.
It seems that what there is no such density, but the "closest" answer to my question is an example of a continuous density that does not asymptotically tend to zero. I am aware of this example from Section 2.7 in the book "Counterexamples in Probability And Statistics", Chapman & Hall/CRC, 1986 by Romano and Siegel. If you know other examples please post them below for future reference.