Linearly Approximatable Functions

The notion of uniform linear approximatability generalizes that of being continuously differentiable. It occurs, e. g., in viscosity solutions of some degenerate partial differential equations. We establish the Holder continuity of uniformly linearly approximatable functions, and we show that functions which are nowhere linearly approximatable form a residual collection of the appropriate Holder space. Finally, we prove an analog of the implicit function theorem applied to level sets.