Namioka's theorem

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In functional analysis, Namioka's theorem is a result concerning the relationship between separate continuity and joint continuity of functions defined on product spaces. Named after mathematician Isaac Namioka, who proved it in his 1974 paper Separate Continuity and Joint Continuity published in the Pacific Journal of Mathematics, the theorem establishes conditions under which a separately continuous function must be jointly continuous on a topologically large subset of its domain.^[1]

Namioka's theorem. Let $X$ be a Čech-complete topological space (such as a complete metric space), $Y$ be a compact Hausdorff space, and $Z$ be a metric space. If $f:X\times Y\rightarrow Z$ is separately continuous, meaning that

for each fixed $x_{0}\in X$ , the function $y\mapsto f(x_{0},y)$ is continuous on $Y$ , and

for each fixed $y_{0}\in Y$ , the function $x\mapsto f(x,y_{0})$ is continuous on $X$ ,
then there exists a dense $G_{\delta }$ -subset $G$ of $X$ such that $f$ is jointly continuous at each point of $G\times Y$ .^[1]^[2]

Namioka's theorem can be equivalently stated in terms of the set $C(f)$ of points where $f$ is continuous, stating that the projection of $C(f)$ onto $X$ contains a dense $G_{\delta }$ subset of $X$ .^[1]^[2]

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