In this paper a full totalization is presented of the Kurzweil-Henstock integral in the multidimensional space. A residual function of the total Kurzweil-Henstock primitive is defined.

We consider and study several weak formulations of the Hessian determinant, arising by formal integration by parts. Our main concern are their continuity properties. We also compare them with the Hessian measure.

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 extend the open mapping theorem and inversion theorem of Robinson for convex multivalued mappings to γ-paraconvex multivalued mappings. Some questions posed by Rolewicz are also investigated. Our results are applied to obtain a generalization of the Farkas lemma for γ-paraconvex multivalued mappings.

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.