Given topological spaces X_1,X_2,…,X_n,Y, consider a multivariable function f:∏^n_(i=1) Xi->Y such that for any (x_1,x_2,…,x_n) in ∏^n_(i=1)X_i, the functions in the family {x->f(x_1,…,x_(i−1),x,x_(i+1),…,x_n)}^n_(i=1) are all continuous. Must f itself be continuous?

jorgejasso85xvx 2022-11-19 Answered
Given topological spaces X 1 , X 2 , , X n , Y, consider a multivariable function f : i = 1 n X i Y such that for any ( x 1 , x 2 , , x n ) i = 1 n X i , the functions in the family { x f ( x 1 , , x i 1 , x , x i + 1 , , x n ) } i = 1 n are all continuous. Must f itself be continuous?
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Answers (1)

luluna81mxmbk
Answered 2022-11-20 Author has 17 answers
The answer is "Yes" if you give the product X i the sliceю
The answer is "No" if you give the product X i the usual product topology. In this case, f is "separately continuous" but not necessarily continuous. The standard example is the function f : R × R R defined by f ( x , y ) = 2 x y x 2 + y 2 for ( x , y ) ( 0 , 0 ) and f ( 0 , 0 ) = ( 0 , 0 ). This function is continuous everywhere except ( 0 , 0 ) but is continuous in each variable.
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