# prove that every continuous function is separately continuous

• November 21st 2011, 04:02 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, 08:34 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, 10:40 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, 07:42 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, 06:29 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, 08:36 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, 04:57 AM
wopashui
Re: prove that every continuous function is separately continuous
ic, but how does this help for the proof?