On the special norm and completeness of spaces of continuous functions of several variables with Lipschitz–Hölder type constraints

Let $X_0\subseteq\mathbb R^n$ be a nonempty open set and $X_0\subseteq X\subseteq\overline X_0$. We admit that the set $X_0$ is unbounded and/or has a countable number of connected components. In this paper, we study some spaces of functions $f\colon X\to\mathbb R$ endowed with a special norm $\|\cdot\|$. The definition of the norm involves an $n$-dimensional vector $(\Delta x)^{-1}\Delta f$, which is an analogue of the relation $\frac{\Delta f}{\Delta x}$ generating the concept of the derivative of a function of one variable. The vector $(\Delta x)^{-1}\Delta f$ can be associated with the vector $\mathrm{grad}\,f(\cdot)$. The invertible matrix $\Delta x$ of order $n$ consists of special increments of the argument $x\in \mathbb R^n$, and the vector $\Delta f$ consists of special increments of the function $f$. A number of properties of the vector $(\Delta x)^{-1}\Delta f$ is proved, and an exact formula for its Euclidean norm is obtained. We prove the completeness with respect to a special norm $\|\cdot\|$ of the space $\mathcal G(X)$ consisting of continuous bounded functions $f\colon X\to\mathbb R$ and having additional restrictions of the Lipschitz–Hölder type. Such functions play an important role in solving mathematical physics problems. A number of important subspaces of the space $\mathcal G(X)$ is investigated. It is proved that two of them are Banach, and one of them, for $n=1$ and under certain conditions, is the closure of the space of piecewise linear functions $f\colon X\to\mathbb R$.