(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let M, N be two metric spaces. For f: M --> N, define the function on M,

graph(f) = {(x,f(x)) [itex]\in[/itex]MxN: x[itex]\in[/itex]M}

show f continuous => graph(f) is closed in MxN

2. Relevant equations

3. The attempt at a solution

I can't figure out what method to use.

I have written out many equivalent defintions of continuity and closed sets, so far I know I want to show that.. basically since f is continuous, we have convergent sequences mapped to convergent sequences, ie xn -> x, fn -> f

And I need to show that (x,f(x)) [itex]\in[/itex] graph(f)

It's either extremely trivial, because from the definition of continuity we see that all the limits belong to that subset..

But anyways, I don't think I can say that or make that a proof by just stating the obvious..

So I tried creating a neighbourhood and showing that M\graph(f) is open.

Assume f is continuous. Then for all x in M, for all ε, there exists δ=θ/2 s/t dM(x,y)<δ => dN(f(x),f(y))<ε

So consider B(y,ε'), choose ε' = δ = θ/2.

For any z in B(y,ε')

d(z,x) ≤ d(z,y) + d(x,y) < ε' + δ = θ

Then d(z,x) < θ and z is in M\graph(f).....

This is all probably very wrong.. I didn't use anywhere the definition of graph(f) which worries me.

eek.. Help?

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# Homework Help: Function continuous, then a subset is closed

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