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?
Continuity of discrete valued function
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continuity
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2The 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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0It 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