Subadditive bijections on the positive reals

Abstract

We show that there exists a subadditive bijection $f:(0,\infty )\to (0,\infty )$ such that ${\mathrm{liminf}}_{x\to 0}f(x)=0$ and ${\mathrm{limsup}}_{x\to 0}f(x)=1$. This answers a question by Matkowski and Swiatkowski in [Proc. Amer. Math. Soc. 119 (1993), 187--197].