Nonsmooth analysis refers to differential analysis in the absence of differentiability. It can be regarded as a subfield of that vast subject known as
nonlinear analysis. While nonsmooth analysis has classical roots (we claim
to have traced its lineage back to Dini), it is only in the last decades that
the subject has grown rapidly. To the point, in fact, that further development has sometimes appeared in danger of being stymied, due to the
plethora of definitions and unclearly related theories.
One reason for the growth of the subject has been, without a doubt, the
recognition that nondifferentiable phenomena are more widespread, and
play a more important role, than had been thought. Philosophically at
least, this is in keeping with the coming to the fore of several other types
of irregular and nonlinear behavior: catastrophes, fractals, and chaos.
In recent years, nonsmooth analysis has come to play a role in functional
analysis, optimization, optimal design, mechanics and plasticity, differential equations (as in the theory of viscosity solutions), control theory, and,
increasingly, in analysis generally (critical point theory, inequalities, fixed
point theory, variational methods ...). In the long run, we expect its methods and basic constructs to be viewed as a natural part of differential
analysis