Is there a real-valued function which is discontinuous everywhere, but which has a limit at every point in it's domain?(adsbygoogle = window.adsbygoogle || []).push({});

My intuition is that this couldn't occur because, if the limit exists at some x, then it must become "increasingly continuous" in the vicinity of x (otherwise we could find sequences to that point with different limits under that function). Then, we could perhaps conclude that f must be continuous in some neighborhood of x. This is of course not a formal proof, though, and I haven't been able to formalize it, hence my curiosity.

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# Discontinuity and limits of functions

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