On some continuity properties of interval-valued function

This paper aims to study some continuity properties of interval-valued function. Shokoohi et al. [Some fundamental theorems in the interval-valued functions and time scales, J. Mahani Math. Res. 15(1) (2025) 281–302] derived Bolzano’s and intermediate value theorems for interval-valued function using the Bolzano’s theorem for real-valued continuous function. However, this approach is trivial and has no deep research merit. In this paper, a Bolzano-type theorem, gH-extreme value theorem, and generalized intermediate value property for an IVCF are derived independently without using the existing usual Bolzano’s theorem. Then, the same results are studied for a particular class of IVCF and further, the usual Bolzano’s theorem, intermediate value property are proved as the particular cases of the findings of this work. Finally, all the findings are illustrated with the help of a number of examples.