# Concrete Mathematics - Confusion regarding use of Intermediate Value Theorem

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:

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?

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