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Define $$A=\left\{\frac1{n+1}: n\in\mathbb N\right\}\subset\mathbb R\\ B=A\cup\{0\}\subset\mathbb R$$ Is any function $f: A\to \mathbb R$ continuous? And is any function $g:B\to\mathbb R$ continuous?

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    Since $A$ has discrete topology, any function $f$ is continuous. For $g$ to be continuous, one only checks the continuity of $g$ at point 0.2012-05-22

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