Separately continuous functions in a new sense are continuous. (English) Zbl 0967.26010

It is known that the separate continuity of a function of several variables does not imply its continuity. Various conditions may be imposed on curves passing through a point \(x^0\) along which a function is continuous without being continuous at \(x^0\). On the other hand, Rosenthal proved that the continuity of a function along all convex differentiable curves through \(x^0\) implies the ordinary continuity of the function at \(x^0\). The article is devoted to conditions on individual variables implying ordinary continuity. An example: A function \(f\) defined in a neighborhood \(U(x^0)\) will be called continuous in the strong sense at \(x^0\) with respect to the variable \(x_k\) if \[ \lim _{x\rightarrow x^0} [ f(x_1,\ldots ,x_{k-1}, x_k, x_{k+1},\ldots ,x_n) - f(x_1,\ldots ,x_{k-1}, x_k^0, x_{k+1},\ldots ,x_n) ]=0\;. \] It is proved that a function \(f\) is continuous at a point \(x^0\) iff it is separately continuous in the strong sense at \(x^0\). Another type of so-called separate continuity in the angular sense is introduced. This notion is shown to be equivalent to separate continuity in the strong sense.


26B05 Continuity and differentiation questions