A New Differential Formalism for Interval-Valued Functions and Its Potential Use in Detecting 1-D Landscape Features

In many practical problems, it is important to know the slope (derivative) dy=dx of one quantity y with respect to some other quantity x. For example, different 1-D landscape features can be characterized by different values of the derivative dy=dx, where y is an altitude, and x is a horizontal coordinate. In practice, we often know the values of y(x) for different x with interval uncertainty. How can we then find the set of possible values of the slope? In this paper, we formulate this problem of differentiating interval-values functions in precise terms, and we describe an (asymptotically) optimal algorithm for computing the corresponding derivative.