## Thomas' Calculus 13th Edition

$\lfloor x \rfloor = \lceil x \rceil$ is true when $x\in N$ Note that in this case, the letter N denotes all integers.
$\lfloor x \rfloor = \lceil x \rceil$ is only true for integers. For any real number between two whole numbers, $\lfloor x \rfloor$ and $\lceil x \rceil$ can't be the same number.