A further extension of interval arithmetic allows better models of ``bumpy'' functions. A function is -bumpy if it is partial or discontinuous. In general, a function g is -bumpy if a better model exists for g in than in . Bumpy functions are formally defined in section . This extension also allows for natural models of multi-functions; multi-functions are functions that may return multiple results, such as .
Throughout this section, let denote , for a fixed choice of f. Each number in is specified as a set of numbers from :
A real number x is contained in interval j if x is contained by any member of j:
|Jeff Tupper||March 1996|