A New Fine-Computability of Functions on [0, 1)

In Fine-computable theory introduced by T. Mori et. al, Fine-computability of functions is known to be equivalent to $(\rho_F, \rho)$-computable, and the Fine-integrability is equivalent to ([\rho_F\rightarrow \rho], \rho)-computable. By introducing the Fine-metric of [0,+\infty), we investigate a new Fine-computability of (non-negative) functions, called Fine$^*$-computable, which is (\rho_F, \rho_F)-computable. In the sense of Fine$^*$-computability, some relations in the classical computable analysis are still remained.