The well-known Men’shov and Gehring-Lehto theorems on the differentiability of topological mappings of plane domains are generalized to the case of continuous open mappings of many-dimensional domains.

We prove that the Rieffel sharpness condition for a Banach space E is necessary and sufficient for an arbitrary Lipschitz function f: [a, b]→E to be differentiable almost everywhere on a segment [a, b]. We establish that, in the case where the sharpness condition is not satisfied, the major part (in the category sense) of Lipschitz functions has no derivatives at any point of the segment [a, b].

We prove theorems analogous to the local maximum principle and theorems on relation between limit sets, which can be used for studying singular sets.

The theorem on pseudoanalyticity of continuous functions with constant σ-extension is proved; this is an analog of the well known results due to Bohr, Rademacher, Men'shov, and Trokhimchuk concerning the analyticity of functions with constant extension.

The following theorem is proved: Every continuous function satisfying the conditionK′ σ is pseudo-analytic. The conditionK′ σ is a generalization of the Men'shov condition, well known in the theory of analytic functions.

The notion of the sets of σ-monogeneity for continuous functions is introduced which makes it possible to study pseudo-analytic properties of these functions. The theorem on the structure of these sets is proved.

New conditions for the C-differentiability of IR-differentiable mappings are obtained and a theorem on the holomorphicity of the mappings that have finite dilatation along cones is proved.

For mappings of infinite-dimensional Hilbert space domains, the concepts of derived operators are introduced, their parametric representation is obtained, and a criterion of coincidence of operators along two subspaces and a holomorphy theorem are proved.

This article considers locally Lipshitzian mappings of domains of infinite-dimensional Hilbert spaces and proves a theorem on holomorphicity of such mappings that satisfy the conditions of C-differentiability.

The concepts of dilation and rotation operators are introduced for maps of domains of infinite-dimensional Hibert spaces. Sufficient conditions for ℂ -differentiability along subspaces are established and theorems on operator conditions for IR-differentiable maps to be monogenic and holomorphic are proved.