The subdifferentiability properties of typical functions in \(C[0,1]\)
Let \(C\) denote the Banach space of continuous real-valued functions on \([0, 1]\). In the paper, the differentiability behavior of typical functions of \(C\) is described, where ``typical'' means all functions, except those in some meager (i.e., first category) subset of \(C\). Naturally, a typical function in such sense is nondifferentiable on \([0,1]\) and it can be shown that for the (upper/lower, right/left) Dini derivatives it is \(f^+(x)= f^-(x) =\infty\), \(f_+ (x)= f_-(x) =-\infty\). In the main theorem of the paper, the author extends this result. Regarding the Jeyakumar convexification \(\partial^*f(x)\), the Dini subdifferential \(\partial^-f(x)\), the Clarke subdifferential \(\partial_cf(x)\) and the Ioffe-Mordukhovich-Kruger approximate subdifferential \(\partial_\alpha f(x)\), respectively, it is shown that there exists a residual set of functions \(f\in C\) such that (1) the set \(E(f)= \{x\in (0,1) \mid f^+ (x)=f^-(x)= +\infty\), \(f_+(x) =f_-(x)= -\infty\}\) is residual in \((0,1)\) and has a full Lebesgue measure; (2) for all \(x\in E(f)\), every closed set in \(\mathbb{R}\) may be chosen as \(\partial^* f(x)\); (3) for all \(x\in E(f)\) it is \(\partial_-f(x) =\emptyset\); (4) for all \(x\in E[0,1]\) it is \(\partial_\alpha f(x)= \partial_cf(x) =\mathbb{R}\). Hence, a typical function turns out to be pathological in a certain sense. At the end of the paper some examples are presented using the well-known nowhere differentiable Weierstrass and van der Waerden functions. Finally, it is pointed out that by the operation \(g(x,y): =f(x) +f(y)\) the results can be extended to the two-dimensional case.
Ioffe-Mordukhovich-Kruger approximate subdifferential
scientific article; zbMATH DE number 1130522
scientific article