Let $f,g$ be continuous on $\mathbb{R}→\mathbb{R}$ and $f(r)=g(r)$ if $r∈\mathbb{Q}$, is it true that $f(x)=g(x)$ for all $x∈\mathbb{R}$
I think it should be true, but I have no idea how to prove it, anyone could help me? Thanks.
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2Take $x\notin \Bbb Q$, and $r_n\in\Bbb Q$ such that $r_n\to x$. Then $f(r_n)=g(r_n)\to ??$ – Pedro Dec 05 '14 at 00:29
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the set ${x:f(x)=g(x)}$ is closed, and since it contains $\Bbb Q$ it is also dense, hence it must be equal to $\Bbb R$. – Mirko Dec 05 '14 at 03:27