Say I have a function F(x,y)=(f(x),g(y)), F:X×Y→X'×Y'. Is there a theorem that says if f:X→X' and g:Y→Y' are continuous then F(x,y) is continuous. I've proved it, or at least I think I have, but I'd like to know for sure whether or not I'm right.(adsbygoogle = window.adsbygoogle || []).push({});

I know that its not necessarily true that a function defined on a product space is continuous even if it is continuous in each variable separately. But it seems as though since the function I defined above does not interact x and y, there may be some different rules.

Also, if anyone knows for sure that this is not true, that would be useful information as well.

Thanks.

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# Looking for a Theorem of Continuous Functions

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