  In this paper, we introduce and investigate a class P of continuous and
periodic functions on R. The class P is defined so that second-order central
differences of a function satisfy some concavity-type estimate. Although this
definition seems to be independent of nowhere differentiable character, it
turns out that each function in P is nowhere differentiable. The class P
naturally appear from both a geometrical viewpoint and an analytic viewpoint.
In fact, we prove that a function belongs to P if and only if some geometrical
inequality holds for a family of parabolas with vertexes on this function. As
its application, we study the behavior of the Hamilton Jacobi flow starting
from a function in P. A connection between P and some functional series is also
investigated. In terms of second-order central differences, we give a necessary
and sufficient condition so that a function given by the series belongs to P.
This enables us to construct a large number of examples of functions in P
through an explicit formula.
