Metric space and calculus of Type-2 interval valued functions

This paper attempts an extensive study on metric space and calculus under Type 2 interval uncertainty. Type 2 interval generalizes interval uncertainty considering both ends of the interval number to be imprecise. Type 2 interval philosophy was introduced in the literature with optimization perspectives. We prioritize the study of Type 2 interval-ruled dynamical systems. The concerns necessitate an extensive introduction of metric space and calculus for Type 2 interval-valued functions. We investigate several fundamental properties of metric space in the contemporary of Type 2 interval setting. After significant findings in differential calculus using generalized Hukuhara difference of Type 2 interval numbers, a detailed and novel manifestation of integral calculus including Riemann and Lebesgue senses is also done in this paper. We also provide hints for possible mathematical modelings of real-world scenarios using Type 2 interval-ruled uncertain decision realm.