This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F\left(x\right):=F̃x^{(2n+1)}$ where $F̃:X^{2n+1}\to Y$ is a continuous (2n+1)-linear operator.

We prove a uniform version of the converse Taylor theorem in infinite-dimensional spaces with an explicit relation between the moduli of continuity for mappings on a general open domain. We show that if the domain is convex and bounded, then we can extend the estimate up to the boundary.

We show that a $C^k$-smooth mapping on an open subset of ${ℝ}^n$, $k\in \mathbb{N}\cup \{0,\infty \}$, can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.