On the module of continuity of mappings with an $s$-averaged characteristic

We continue studying analytical properties of non-homeomorphic mappings with an $s$-averaged characteristic. O. Martio proposed the theory of $\mathcal{Q}$-homeomorphisms (2001). The concept of $\mathcal{Q}$-homeomorphisms was extended to maps with branching (2004). In this paper, we study analytical properties of non-homeomorphic mappings with an $s$-averaged characteristic and consider the question of continuity of mappings with an $s$-averaged characteristic. By the well-known Sobolev theorem, a function of class $W^1_{s,loc}(R^n)$ for is equivalent to a continuous function. This property does not hold when $s<n$. The authors presented such example for mappings with an $s$-averaged characteristic in 2016.
In this paper, we generalize the result obtained earlier to a more general class of mappings with an $s$-averaged characteristic. Relevant examples are built. The purpose of this paper is to indicate the necessary conditions under which mappings from classes and subclasses of mappings with an $s$-averaged characteristic $1<s<n$ will be continuous. Here, $n$ is the dimension of the space, and $s$ is the averaging parameter. We proved a theorem in which we obtain necessary conditions for the continuity of such mappings that are with the abovementioned $s$. Earlier, such a result was obtained for functions of the class $W^1_{s,loc}(R^n)$. The theorem is an analogue of the Mori lemma.