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