Continuity by difference schemes

This approach uses the definition of continuity, that a function f(x) is continuous at an abscissa x if 

The sequence of 6x used for taking this limit is conveniently the sequence of polygon edges at successive refinements of the original polygon. 

The question is whether we can bound the values of f(x + Sx) — f(x) in terms of the original control points, and the answer is yes , using the neat idea of a difference scheme, which relates the first differences of the new polygon to the first differences of the old. 

Suppose that a binary scheme, S, has a -transform of S(z) and the old polygon is Po(z2). Then the new polygon is given by 

Thus if S(z) is divisible by 1 + z (and it always is if each of the stencils sums to 1) we can take the quotient as a scheme which relates first differences of P to first differences of Pq. Call this scheme D(z). 

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