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In Chapter 3 of Concrete Mathematics, Second Edition by Knuth, Graham and Patashnik, the authors prove a property of the ceiling (or floor) function. Here it is:

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In lines 4 to 3 (from the bottom), I believe the Intermediate Value Theorem is used on the interval $[x,\lceil{x}\rceil)$. However, for that, we require the inequality $f(x) \leq \lceil{f(x)}\rceil \leq f(\lceil{x}\rceil)$. But we have $f(x) \leq \lceil{f(x)}\rceil < \lceil{f(\lceil{x}\rceil)}\rceil$.

How can we then apply the IVT?

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  • $\begingroup$ I now realized that this question might be a duplicate: math.stackexchange.com/a/1859233/592479 $\endgroup$ – DS2830 Nov 21 '19 at 13:26
  • $\begingroup$ @Maruo The special property holds for $f$ not for the ceiling of $f$. $\endgroup$ – DS2830 Nov 21 '19 at 13:28

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