Other frequently used and useful functions are the lower integer part and the upper integer part of a real number.
The lower integer part of a real number $x$ is defined as the gratest integer number which is smaller than or equal to $x$ and we denote it as $\lfloor x \rfloor$. Similarly, the upper integer part of a real number $x$ is defined as the smallest integer number that is greater than or equal to $x$ and we denote it as $\lceil x \rceil$. Thus, we could explicitly write (we just express symbolically what we have written in the previous sentences):
The domain of the upper and lower integer part is the whole real line $\mathbb{R}$. Trivially, according to its definition, we are able to costruct $\lfloor x \rfloor$, resp. $\lceil x \rceil$, for every real number $x$. The set of values of these functions is then given by the set of all the integers, i.e. $\mathbb{Z}$. The graphs of the upper and lower integer parts are shown in the figure 4.2.