A mathematical smooth function means that the function has continuous
derivatives to a certain degree C(k). We call it a k-smooth function or a
smooth function if k can grow infinitively. Based on quantum physics, there is
no such smooth surface in the real world on a very small scale. However, we do
have a concept of smooth surfaces in practice since we always compare whether
one surface is smoother than another one. This paper deals with the possible
definitions of "natural" smoothness and their relationship to the original
mathematical definition of smooth functions. The motivation of giving the
definition of a smooth function is to study smooth extensions for practical
applications.

We observe this problem from two directions: From discrete to continuous, we
suggest considering both micro smooth, the refinement of a smoothed function,
and macro smooth, the best approximation using existing discrete space. (For
two-dimensional or higher dimensional cases, we can use Hessian matrices.) From
continuous to discrete, we suggest a new definition of natural smooth, it uses
a scan from down scaling to up scaling to obtain the a ratio for sign changes
by ignoring zero to represent the smoothness. For differentiable functions,
mathematical smoothness does not mean a "good looking" smooth for a sampled set
in discrete space. Finally, we discuss the Lipschitz continuity for defining
the smoothness, which will be called discrete smoothness. This paper gives
philosophical consideration of smoothness for practical problems, rather than a
mathematical deduction or reduction, even though our inferences are based on
solid mathematics.