The Range of a Continuous Functional Under Set-Valued Uncertainty Is Always an Interval

One of the main problems of interval computations is computing the range of a given function on a given multi-D interval (box). It is known that the range of a continuous function on a box is always an interval. However, if, instead of a box, we consider the range over a subset of this box, the range is, in general, no longer an interval. In some practical situations, we are interested in computing the range of a functional over a function defined with interval (or, more general, set-valued) uncertainty. At first glance, it may seem that under a non-interval set-valued uncertainty, the range of the functional may be different from an interval. However, somewhat surprisingly, we show that for continuous functionals, this range is always an interval.