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The question is the following: is it possible for a (say) real function to be continuous at a certain pointto its domain, and be discontinuous in some neighborhood of that point?internal

I am not talking about a function defined at a single point or things like that, but of a function defined on the entire [itex]\mathbb{R}[/itex] (or some interval in it, whatever).

Now, i have also came up with an answer: a function [itex]f[/itex] defined as [itex]f(0)=0[/itex], [itex]f(x)=x[/itex] for every rational [itex]x[/itex], and [itex]f(x) = 2x[/itex] for every irrational [itex]x[/itex]. Such a function would be (seems to me) continuous at [itex]x=0[/itex] and discontinuous for any other [itex]x[/itex]. I am not completely certain of this, though, and for that reason i would like some feedback on this.

I am also asking this question because strangely enough I have never heard of the concept of an isolated continuity point, while for example the "opposite" (that of an isolated singularity) is quite common, and I would like to know if it's just me or if it is just a "useless" pathological concept.

Thank you in advance.

Bye

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# Isolated continuity point

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