Nowhere Weak Differentiability of the Pettis Integral

Lineability of non-differentiable Pettis primitives

Let X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0; 1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND.

Locally Lipschitz Functions and Bornological Derivatives

We study the relationships between Gateaux, weak Hadamard and Frechet differentiability and their bornologies for Lipschitz and for convex functions. In particular, Frechet and weak Hadamard differentiabily coincide for all Lipschitz functions if and only if the space is reflexive (an earlier paper of the first two authors shows that these two notions of differentiability coincide for continuous convex functions if and only if the space does not contain a copy of $\ell_1$). We also examine when Gateaux and weak Hadamard differentiability coincide for continuous convex functions. For instance, …

Nowhere differentiable functions of analytic type on products of finitely connected planar domains

Using Laurent decomposition and Mergelyan's theorem combined with Baire's category theorem, we prove generic nowhere differentiability on the distinguished boundary of functions of analytic type on products of planar domains bounded by finitely many disjoint Jordan curves. The parametrization of the boundaries are those induced by the natural Riemann maps.

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

The aim of this expository article is to present recent developments in the centuries old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include, among others, the $D^n$-$C^n$ interpolation theorem: {\em For every $n$-times differentiable $f\colon\R\to\R$ and perfect $P\subset \R$ there is a $C^n$ function $g\colon\R\to\R$ such that $f\restriction P$ and $g\restriction P$ agree on an uncountable set} and an example of a differentiable function $F\colon\R\to\R$ (which can be nowhere monotone) and of compac…

具有图维数、霍尔德正则性和分数可微性的代数结构

Algebraic structures featuring graph dimensions, Hölder regularity, and fractional differentiability

研究了具有极端不规则性的连续函数的代数普遍性，证明了某些函数集的强c-可代数性，并探讨了霍尔德指数与分数可微性之间的关系。

Further Notes on a Family of Continuous, Non-differentiable Functions

We examine a parameterized family of functions F_a, all of which are continuous and some of which are nowhere or almost nowhere differentiable, we explore the behavior of F'_a and F"_a almost everywhere for different values of a, focusing on specific questions regarding F_a's differentiability for certain a, and we calculate the Hausdorff dimension of the graphs of all F_a.