We observe that each set from the system $\widetilde{𝒜}$ (or even $\widetilde{𝒞}$) is $\Gamma$-null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ${ℝ}^n$ is $\sigma$-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex...

Let $I$ be an open interval, $X$ a topological space and $Y$ a metric space. Some local conditions implying continuity and quasicontinuity of almost all sections $x\to f(t,x)$ of a function $f:I\times X\to Y$ are shown.

It is proved that the following conditions are equivalent: (a) f is an almost everywhere continuous function on ${ℝ}^m$; (b) f = g + h, where g,h are strongly quasicontinuous on ${ℝ}^m$; (c) f = c + gh, where c ∈ ℝ and g,h are strongly quasicontinuous on ${ℝ}^m$.

We determine conditions in order that a differentiable function be approximable from above by analytic functions, being left invariate on a fixed analytic subset which is a locally complete intersection.

The classical Denjoy-Young-Saks theorem gives a relation, here termed the Denjoy property, between the Dini derivatives of an arbitrary one-variable function that holds almost everywhere. Concerning the possible generalizations to higher dimensions, A. S. Besicovitch proved the following: there exists a continuous function of two variables such that at each point of a set of positive measure there exist continuum many directions, in each of which one Dini derivative is infinite and the other...

We characterize sets of non-differentiability points of convex functions on ${ℝ}^n$. This completes (in ${ℝ}^n$) the result by Zajíček [2] which gives a characterization of the magnitude of these sets.

We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a $C^p$-smooth bump b:X → ℝ such that $\left|\right|b^{\left(p\right)}{\left|\right|}_{\infty }$ is finite, then any connected open subset of X* containing 0 is the range of the derivative of a $C^p$-smooth bump. We also study the finite-dimensional case which is quite different. Finally,...

A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f\left(Ū\right)\subseteq \overline{f\left(U\right)}$ for any open connected set U ⊆ X. We prove that if X is a locally connected hereditarily Baire space and Y is a T₁-space then an $F_{\sigma }$-measurable mapping f: X → Y is weakly Gibson if and only if for any connected set C ⊆ X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson $F_{\sigma }$-measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph.

We establish the embedding of the critical Sobolev-Lorentz-Zygmund space $H_{p,q,\lambda ₁,...,\lambda ₘ}^{n/p}\left(ℝⁿ\right)$ into the generalized Morrey space ${ℳ}_{\Phi ,r}\left(ℝⁿ\right)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H_{p,q,\lambda ₁,...,\lambda ₘ}^{n/p+1}\left(ℝⁿ\right)$. O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.