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Suppose say $f:\{0,1\}\to \{1,2\}$ is $f$ continuous? Say $f(0)=2,f(1)=1$ I know the definition of continuous function. In my point of view, i think it is continuous as we can simply take $\epsilon=\delta$, so for any $x\in \{0,1\},|x-1|<\delta \implies |f(x)-f(1)|<\epsilon$. Is it correct?

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    @Nameless intuitively this $f$ only contains 2 isolated points so i feel it is not continuous but from the definition, if I understood correctly, it seems to be continuous so that's why i doubt about whether $f$ is a continuous functions2012-11-18
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    @Nameless if $\epsilon<1$ then we know that $f(x)=f(1)$ and $|x-1|=|1-1|=0<\delta|$2012-11-18
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    @Mathematics A function is always continuous at isolated points. That's why the $\epsilon-\delta$ definition is a little more general to the limit definition of continuity2012-11-18

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