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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. –  Xiaochuan May 22 '12 at 14:14

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Since $A$ is discrete, every subset of $A$ is open so any function $f:A\to\mathbb R$ is continuous (the preimage of any open subset of $\mathbb R$ must be open). However, $B$ is not discrete, so there is some function $f:B\to \mathbb R$ which is not continuous. Can you show that the function $f$ defined by $f(x)=1/x$ for $x\in A$ and $f(0)=0$ is not continuous at $0$?

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