A Note about Differentiability of Maps Defined on Convex Subsets of Banach Spaces That May Be Nowhere Dense*

In connection with continuum mechanics there are physically meaningful choices of infinite-dimensional Banach spaces such that the domain of constitutive maps is nowhere dense in them, as V. J. Mizel and C.-C. Wang Arch. Rational Mech. Anal. 23, 1996, 124]134. pointed out. Thus the usual differential calculus on open sets cannot be applied there. Here we give a differentiability notion for maps f defined on any convex subset of a Banach space that may be nowhere dense. When the domain of f is open, this notion coincides with the usual one. We give the definitions and prove the theorems related to first and higher order derivatives of f.