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I've already seen this link - Montone Function only has Jump Discontinuities - but I'm having trouble 'filling in the blanks' in the proof.

Would anyone be able to walk me through a complete proof of this? I've been struggling for several hours now..

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Without loss of generality assume it is non-decreasing.

At the point $x=a$ the $\lim_{x\to a-}f(x)$ is bounded by $f(a)$ from above and $f$ increases in $(-\infty,a)$. Therefore the lateral limit from the left exists. On the other hand $\lim_{x\to a+}f(x)$ is bounded from below by $f(a)$ and $f$ increases in $(a,+\infty)$. Therefore the lateral limit from the right exists.

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