Thread: prove that every continuous function is separately continuous

1. 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.

2. Re: prove that every continuous function is separately continuous

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.

3. Re: prove that every continuous function is separately continuous

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?

4. Re: prove that every continuous function is separately continuous

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.

5. Re: prove that every continuous function is separately continuous

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}?

6. Re: prove that every continuous function is separately continuous

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.

7. Re: prove that every continuous function is separately continuous

ic, but how does this help for the proof?