My problem is this. Let [tex]f:\mathbb{R}^{2}\longrightarrow \mathbb{R}^{2}[/tex] be a continuous function that satifies that [tex]\forall q\in\mathbb{Q}\times\mathbb{Q}[/tex] we have [tex]f(q)=q[/tex]. Proof that [tex]\forall x\in\mathbb{R}^{2}[/tex] we have [tex]f(x)=x[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

I have worked out that because it is continuous, [tex]f[/tex] satisfies that

[tex]\forall \epsilon>0\exists\delta>0\mid \forall x\in B_{\delta}(a)\longleftrightarrow f(x)\in B_{\epsilon}(f(a))[/tex]

and then [tex]\forall q\in\mathbb{Q}\times\mathbb{Q}[/tex] we have

[tex]\forall \epsilon>0\exists\delta>0\mid \forall x\in B_{\delta}(q)\longleftrightarrow f(x)\in B_{\epsilon}(q)[/tex]

therefore we have to proof that [tex]\forall x'\in\mathbb{R}^{2}[/tex] we have

[tex]\forall \epsilon>0\exists\delta>0\mid \forall x\in B_{\delta}(x')\longleftrightarrow f(x)\in B_{\epsilon}(x')[/tex].

It's obvious that every element of [tex]\mathbb{R}^{2}[/tex] could be approximated by some element of [tex]\mathbb{Q}\times\mathbb{Q}[/tex] or sequence in this. But, how I can link this in an expression to get what I have to proof?

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# Homework Help: Continuity of a function

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