On the limit and continuity of functions of several variables (Q2774463)

The article is devoted to find new conditions on continuity for functions of several variables. It is based on the notion of separately strong continuity defined by the author in a precedent work [Real Anal. Exch. 24, No. 2, 695-702 (1998; Zbl 0967.26010)]. The definition of separately strongly continuous function is the following: A function \(f\) defined in a neighborhood \(U(x^0)\) is called strongly continuous at \(x^0\) with respect to the variable \(x_k\) if NEWLINE\[NEWLINE \lim _{x\rightarrow x^0} [ f(x_1,\ldots ,x_{k-1}, x_k, x_{k+1},\ldots ,x_n) - f(x_1,\ldots ,x_{k-1}, x_k^0, x_{k+1},\ldots ,x_n) ]=0.NEWLINE\]NEWLINE Two main results of the paper are the following: NEWLINENEWLINENEWLINEA function \(f\) is continuous at a point \(x^0\) iff it is separately continuous at \(x^0\) with respect to one variable and has finite limit at \(x^0\). NEWLINENEWLINENEWLINEA function \(f\) is continuous at a point \(x^0\) iff it is separately continuous at \(x^0\) with respect to one variable and separately strongly continuous at \(x^0\) with respect to each of the rest \(n-1\) variables.