1 Introduction
2 Mathematics is not only about computing
3 Basic concepts
4 Elementary functions
4.1 What is it a function?
4.2 The absolute value
4.3 Lower and upper integer part
4.4 Linear function
4.5 Quadratic function
4.6 Polynomial function
4.7 Roots
4.8 Rational function
4.9 Trigonometric functions
4.10 Exponantiation and logarithm
5 Analytical geometry
6 Warning
7 List of the used symbols
Index
Bibliography

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):

\begin{equation*}
\begin{aligned}
\lfloor x \rfloor & \href{The entry on the right repeats what was written in the previous paragraph by using only symbols. Here we have the set of all integers \(m\), that are smaller than or equal to \(x\) and we take the largest element (maximum) of such a set.}{\class{mathpopup bg-info-subtle}{=}}
\max \{ m \in \Z \mid m \leq x \}, \\
\lceil x \rceil & \href{The entry on the right repeats what was written in the previous paragraph by using only symbols. Here we have the set of all integers \(m\), which are grster than or equal to \(x\) and we take the smallest set of such a set.}{\class{mathpopup bg-info-subtle}{=}}
\min \{ m \in \Z \mid m \geq x \}.
\end{aligned}
\end{equation*}

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.