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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.

##### MSC:
 26B05 Continuity and differentiation questions
##### Keywords:
separate continuity in the strong sense