jorgejasso85xvx

2022-11-19

Given topological spaces ${X}_{1},{X}_{2},\dots ,{X}_{n},Y$, consider a multivariable function $f:\prod _{i=1}^{n}{X}_{i}\to Y$ such that for any $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\in \prod _{i=1}^{n}{X}_{i}$, the functions in the family $\left\{x↦f\left({x}_{1},\dots ,{x}_{i-1},x,{x}_{i+1},\dots ,{x}_{n}\right){\right\}}_{i=1}^{n}$ are all continuous. Must $f$ itself be continuous?

luluna81mxmbk

The answer is "Yes" if you give the product $\prod {X}_{i}$ the sliceю
The answer is "No" if you give the product $\prod {X}_{i}$ the usual product topology. In this case, $f$ is "separately continuous" but not necessarily continuous. The standard example is the function $f:\mathbb{R}×\mathbb{R}\to \mathbb{R}$ defined by $f\left(x,y\right)=\frac{2xy}{{x}^{2}+{y}^{2}}$ for $\left(x,y\right)\ne \left(0,0\right)$ and $f\left(0,0\right)=\left(0,0\right)$. This function is continuous everywhere except $\left(0,0\right)$ but is continuous in each variable.

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