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Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a function (not necessarily continuous) from the real line to the real line,

how to prove that the set $V\triangleq \{ a\in \mathbb{R}| \overline{\lim}_{x\mapsto a+} f(x)\neq \overline{\lim}_{x\mapsto a-}f(x) \}$ is countable?

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    Related: http://math.stackexchange.com/questions/84870/how-to-show-that-a-set-of-discontinuous-points-of-an-increasing-function-is-at-most-countable2011-11-25
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    could you give some explanation, how to define a monotonic function from $f(x)$ to reduce this question to the link you mentioned?2011-11-25
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    I didn't say it was reducible to there, just that it was similar.2011-11-25
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    What is $\overline{\textrm{lim}}$?2011-11-25
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    @user7530, it's an alternative notation for [limsup](http://en.wikipedia.org/wiki/Limit_superior).2011-11-25
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    Another hint. For each pair of rationals $u, investigate how many points $a$ can have $$\overline{\lim_{x\mapsto a+}}f(x) < u < v < \overline{\lim_{x\mapsto a-}}f(x)$$.2011-11-25

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