prove that every continuous function is separately continuous

Let (M,d),(N,p),(Y,q) be metric spaces. A function f:MxN-->Y is called separately continuous if for every and , the function and , given by and , are continuous.

Prove that every continuous function f:MxN-->Y is separately continuous.

the deinition given is confusing, I have not done any separately continuous question before, need some help here.

Re: prove that every continuous function is separately continuous

Quote:

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**wopashui** Let (M,d),(N,p),(Y,q) be metric spaces. A function f:MxN-->Y is called separately continuous if for every

and

, the function

and

, given by

and

, are continuous.

Prove that every continuous function f:MxN-->Y is separately continuous.

the deinition given is confusing, I have not done any separately continuous question before, need some help here.

Prove that you have the obvious embedding is continuous and note then that , etc.

Re: prove that every continuous function is separately continuous

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**Drexel28** Prove that you have the obvious embedding

is continuous and note then that

, etc.

sorry, what is embedding? can you explain a bit more about the proof?

Re: prove that every continuous function is separately continuous

Quote:

Originally Posted by

**wopashui** sorry, what is embedding? can you explain a bit more about the proof?

There are two ways you can think about it. Perhaps the easiest is, despite what I said, to merely note that is just restricted to and restrictions of continuous functions are continuous.

Re: prove that every continuous function is separately continuous

Quote:

Originally Posted by

**Drexel28** There are two ways you can think about it. Perhaps the easiest is, despite what I said, to merely note that

is just

restricted to

and restrictions of continuous functions are continuous.

hmm, what about g(x), is g the same as f restricted to Mx{b}?

Re: prove that every continuous function is separately continuous

Quote:

Originally Posted by

**wopashui** hmm, what about g(x), is g the same as f restricted to Mx{b}?

Indeed. Of course, technically we are talking about the restrictions with the obvious identifications and , of course this identification is via the embedding I previously mentioned.

Re: prove that every continuous function is separately continuous

ic, but how does this help for the proof?