  Numerical solutions of differential equations are usually not smooth
functions. However, they should resemble the smoothness of the corresponding
real solutions in one way or another. In two of our recent papers, a kind of
spacial smoothness indicators was defined and subsequently applied on the {\it
a posteriori} error analysis. Here we prove that the boundedness of those
smoothness indicators is actually a necessary condition for a piecewise
polynomial function to approximate a smooth function with optimal convergence
rate. This should help in validating the error analysis in the related papers.
Moreover, the result of this paper provides an efficient practical method to
detect the loss of convergence rate due to the lack of numerical smoothness,
hence it serves as a criterion for the qualities of many numerical schemes.
