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YCor
YCor
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Additional conditions under which separately continuous implies jointjointly continuous

Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.

I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):X\rightarrow X$ being surjective or injective for every $y\in Y$) which would grant the joint continuity of $f:X\times Y\rightarrow X$.

The strongest result I have found is contained in this article by Namioka and states that in this case there is a dense subset $A\subset Y$ such that $f:X\times A\rightarrow X$ is jointjointly continuous, but this is not what I am looking for.

Additional conditions under which separately continuous implies joint continuous

Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.

I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):X\rightarrow X$ being surjective or injective for every $y\in Y$) which would grant the joint continuity of $f:X\times Y\rightarrow X$.

The strongest result I have found is contained in this article by Namioka and states that in this case there is a dense subset $A\subset Y$ such that $f:X\times A\rightarrow X$ is joint continuous, but this is not what I am looking for.

Additional conditions under which separately continuous implies jointly continuous

Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.

I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):X\rightarrow X$ being surjective or injective for every $y\in Y$) which would grant the joint continuity of $f:X\times Y\rightarrow X$.

The strongest result I have found is contained in this article by Namioka and states that in this case there is a dense subset $A\subset Y$ such that $f:X\times A\rightarrow X$ is jointly continuous, but this is not what I am looking for.

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user493456
user493456
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Additional conditions under which separately continuous implies joint continuous

Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.

I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):X\rightarrow X$ being surjective or injective for every $y\in Y$) which would grant the joint continuity of $f:X\times Y\rightarrow X$.

The strongest result I have found is contained in this article by Namioka and states that in this case there is a dense subset $A\subset Y$ such that $f:X\times A\rightarrow X$ is joint continuous, but this is not what I am looking for.