A pointwise selection principle for maps of several variables via the total joint variation

Given a rectangle in the real Euclidean n -dimensional space and two maps f and g defined on it and taking values in a metric semigroup, we introduce the notion of the total joint variation TV ( f , g ) of these maps. This extends similar notions considered by Hildebrandt (1963) [17], Leonov (1998) [18], Chistyakov (2003, 2005) [5,8] and the authors (2010). We prove the following irregular pointwise selection principle in terms of the total joint variation: if a sequence of maps  { f j } j = 1 ∞  from the rectangle into a metric semigroup is pointwise precompact and  lim sup j , k → ∞  TV ( f j , f k )  is finite, then it admits a pointwise convergent subsequence (whose limit may be a highly irregular, e.g., everywhere discontinuous, map). This result generalizes some recent pointwise selection principles for real functions and maps of several real variables.