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Is the ceiling function continuous when considered as a function from real numbers to integers (with discrete topology), and what is the formal argument for the proof? Do we have general results about that kind of functions?

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    The preimages of the open(!) sets $\{n\}$ aren't open in $\mathbb R$, hence the function isn't continuous.2012-06-19
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    It depends on the topology on $\mathbb{R}$. It is continuous with the lower limit topology on $\mathbb{R}$, but not with the 'usual' topology.2012-06-19

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