Continuity-Forcing for Derivatives in Data Reconstruction

The smooth function reconstruction needs to use derivatives. In 2010, we used the gradually varied derivatives to successfully constructed smooth surfaces for real data. We also briefly explained why the gradually varied derivatives are needed. In the this paper, we present more reasons to enlighten of forcing derivatives to be continuous is necessary. This requirement seems not a must in theory for functions in continuous space, but it is truly important in function reconstruction for real problems. This paper is also to extend the meaning of the methodology for gradually varied derivatives to general purposes by considering forcing calculated derivatives to be "continuous" or gradually varied.