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 (ρ_F, ρ)-computable, and the Fine-integrability is equivalent to ([ρ_F → ρ],ρ)-computable. By introducing the Fine-metric of [0,+∞), we investigate a new Fine-computability of (non-negative) functions, called Fine*-computable, which is (ρ_F, ρ_F)-computable. In the sense of Fine*-computability, some relations in the classical computable analysis are still remained.