On subadditive functions

The main result says that every one-to-one subadditive function $f:(0,\infty ) \to (0,\infty )$ such that ${\lim _{t \to 0}}f(t) = 0$ must be continuous everywhere. A construction of a broad class of discontinuous subadditive bijections of $(0,\infty )$ which are bounded in every vicinity of 0 is given. Moreover, a problem of extension of a subadditive function defined in $(0,\infty )$ to a subadditive even function in $\mathbb{R}$ is considered