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This question already has an answer here:

$f:\mathbb R\rightarrow \mathbb R$ has both a left limit and a right limit at each point of $\mathbb R.$ Then the number of discontinuities of $f$ is what $?$

Now the Greatest Integer Function is one such function and has countable infinite discontinuities.

Any continuous function also has both limits at each point and they are same and number of discontinuity is $0.$

So , is there a function with both limits at each point that has number of discontinuities uncountable $?$ Is that possible $?$

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marked as duplicate by John, Claude Leibovici, user228113, SchrodingersCat, user91500 Dec 30 '15 at 11:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Please see the accepted answer to this $\endgroup$ – John Dec 30 '15 at 8:10
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The number of discontinuities of such an $f $ is at most countable. Indeed, since the lateral limits always exist, every discontinuity is a jump one; and the number of jump discontinuities is at most countable.

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