3.2.23 Binary Functions

Given the binary function ,   let and denote unary functions, for any . These functions are one-dimensional slices of g, for a constant x or constant y. The functions are defined as follows:

We now restrict our attention to grid functions. The function g is a grid function if it is defined over a grid:

where, for any :

A grid function may be classified using the scheme set out for unary functions:

A function fits into a class if it may be extended into a grid function which fits into that class:

With this classification scheme, the function g may be cut into sections where each section fits into a class:

As with unary functions, denotes a preferred sectioning from which covers are formed.

An upper bound for , where , is determined by considering , for all , and then , for a particular . Since , the same produces an upper bound of . Exceptional functions, whether they are partial, discontinuous or bumpy, are handled as before. For g, where :

where . We assume that ; if not we may extend g|D, as was done with unary functions.

Jeff Tupper

March 1996