I posed some time ago this question on MSE, which I am proposing also here since we got no definitive answer.
Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)$ such that $$ \liminf_{x\to 0^+}f(x)=0 \,\,\,\text{ and }\,\,\,\limsup_{x\to 0^+}f(x)=1\,? $$
As it is written in the comments therein, the answer would be affirmative if: (i) "bijection" is replaced with "injection"; (ii) "$\limsup_{x\to 0^+}f(x)=1$" is replaced with "$\limsup_{x\to 0^+}f(x)\neq 0$"; and (iii) "$\liminf_{x\to 0^+}f(x)=0$" is replaced with "$\liminf_{x\to 0^+}f(x)\neq 0$".
