TITLE: Finding all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$
QUESTION [6 upvotes]: Question:

Find all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$.

The notation $\lceil x \rceil$ means: The least integer which is not less than $x$.
My ideas:
Let's say $\lceil x\rceil=y$ is an integer. If we find $y$ then we'd be able to say $y-1 < x < y$. With this, $y \lceil y \lceil y \lceil y \rceil \rceil \rceil = 88 \to y^4 = 88$, which has no integer solutions.
Am I approaching it wrongly? Where do I start?

REPLY [10 votes]: There is no number $x$ for which $f(x)=x[x[x[x]]]$ equals $88$.
$f(x)$ is an increasing function over $\mathbb{R}^+$ and:
$$ f(3)=81,\qquad \lim_{x\to 3^+} f(x) = 120. $$
$f(x)$ is a decreasing function over $(-\infty,-1]$ and:
$$ f(-3) = 81, \qquad \lim_{x\to -3^-}f(x) = 90.$$