prove that every continuous function is separately continuous

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November 21st 2011, 03:02 PM
wopashui

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 $a\in M$ and $b \in M$ , the function $g:N-->Y$ and $h:M-->Y$ , given by $g(y)=f(a,y)$ and $h(x)=f(x,b)$ , 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.
November 21st 2011, 07:34 PM
Drexel28

Re: prove that every continuous function is separately continuous

Quote:

Originally Posted by wopashui

Let (M,d),(N,p),(Y,q) be metric spaces. A function f:MxN-->Y is called separately continuous if for every $a\in M$ and $b \in M$ , the function $g:N-->Y$ and $h:M-->Y$ , given by $g(y)=f(a,y)$ and $h(x)=f(x,b)$ , 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 $\iota:N\to M\times N:h\mapsto (a,y)$ is continuous and note then that $h=f\circ\iota$ , etc.
November 21st 2011, 09:40 PM
wopashui

Re: prove that every continuous function is separately continuous

Quote:

Originally Posted by Drexel28

Prove that you have the obvious embedding $\iota:N\to M\times N:h\mapsto (a,y)$ is continuous and note then that $h=f\circ\iota$ , etc.

sorry, what is embedding? can you explain a bit more about the proof?
November 22nd 2011, 06:42 AM
Drexel28

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 $h$ is just $f$ restricted to $\{a\}\times N$ and restrictions of continuous functions are continuous.
November 22nd 2011, 05:29 PM
wopashui

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 $h$ is just $f$ restricted to $\{a\}\times N$ and restrictions of continuous functions are continuous.

hmm, what about g(x), is g the same as f restricted to Mx{b}?
November 22nd 2011, 07:36 PM
Drexel28

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 $M\approx M\times\{b\}$ and $N\approx \{a\}\times M$ , of course this identification is via the embedding I previously mentioned.
November 23rd 2011, 03:57 AM
wopashui

Re: prove that every continuous function is separately continuous

ic, but how does this help for the proof?