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This paper deals with the new concept of interval whose both the boundsthemselves are also intervals. We name this new type of interval as Type-2interval. Here we have introduced Type-2 interval-valued function and itsproperties. To serve this purpose, we have defined a distance on the set of allType-2 intervals, named as extended Moore distance for Type-2 intervals whichis a metric on the set of all Type-2 intervals. Then we have shown that thespace of Type-2 interval is a complete metric space with respect to extendedMoore distance. Then we have introduced the concept of limit-continuity forType-2 interval-valued function of single variable and also, we have derivedsome elementary properties of this concept. Subsequently, we have presented theidea of generalised Hukuhara difference on the set of Type-2 intervals.Finally, using this difference, we have defined gH-differentiability of Type-2interval-valued function and discussed some of its properties.

On the space of Type-2 interval with limit, continuity and differentiability of Type-2 interval-valued functions