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Integer part and fractional part functions

Part 3 · Functions · Chapter 7 · Lecture notes, volume 3 (PDF) · Exercises, volume 3 (PDF) · Slides (PDF)

1. Integer part and fractional part functions

  • Two functions that are typically encountered when writing algorithms are the integer part (floor) function and the fractional part (or decimal part) function.

Definition 1: of the integer part function

The integer part (floor) function is:

\[\begin{equation} \label{parte_int} f: \mathbb{R} \rightarrow \mathbb{Z}, x \mapsto [x] \qquad ({\rm or~~ x \mapsto \lfloor x \rfloor}) \end{equation}\]

Definition 2: of the upper integer part function

The upper integer part (ceiling) function is:

\[\begin{equation} \label{parte_ceil} f: \mathbb{R} \rightarrow \mathbb{Z}, x \mapsto \lceil x \rceil \end{equation}\]

Definition 3: of the fractional part function

The fractional part function is:

\[\begin{equation} \label{parte_int__2} f: \mathbb{R} \rightarrow [0, 1), x \mapsto (x) \end{equation}\]

  • The integer part function \(f(x)= \lfloor x \rfloor\) and the upper integer part function \(f(x)= \lceil x \rceil\) are non-decreasing in \(\mathbb{R}\), but they are not increasing.

    • They are non-decreasing: let \(x_1 < x_2\). Then \(\lfloor x_1 \rfloor \le x_1 < x_2\) and, since \(\lfloor x_1 \rfloor\) is an integer less than or equal to \(x_2\), while \(\lfloor x_2 \rfloor\) is the largest integer less than or equal to \(x_2\) (see the chapter “Roots, powers, logarithms and modular arithmetic” of Part 2), we have \(\lfloor x_1 \rfloor \le \lfloor x_2 \rfloor\). In the same way \(\lceil x_2 \rceil \ge x_2 > x_1\) and, since \(\lceil x_1 \rceil\) is the smallest integer greater than or equal to \(x_1\), we have \(\lceil x_1 \rceil \le \lceil x_2 \rceil\).

    • They are not increasing: both are constant on every interval between two consecutive integers, for instance

      \[ \left\lfloor \frac{1}{3} \right\rfloor = \left\lfloor \frac{2}{3} \right\rfloor = 0 \qquad {\rm ~~and~~} \qquad \left\lceil \frac{1}{3} \right\rceil = \left\lceil \frac{2}{3} \right\rceil = 1, \]

      hence there exist \(x_1 < x_2\) with \(f(x_1) = f(x_2)\).

  • Graphs of the functions \(f(x)=\lceil x \rceil\) and \(f(x)= \lfloor x \rfloor\) on the interval \([-4,4]\).

    Figure 1

    Figure 2

  • Note that the fractional part is a periodic function with period 1:

    Figure 3

2. Piecewise-defined functions

  • Starting from the elementary functions, new ones can be built by using different analytic definitions on different intervals.

Definition 4: of piecewise-defined function

A function \(f\) whose value \(f (x)\) is computed through different “instructions” depending on the interval in which \(x\) lies is called a piecewise-defined function.

Example 1: Piecewise-defined functions

Consider for example

\[ f(x)= \begin{cases} \ln x & {\rm if~~} x > 1,\\ x^2 & {\rm if~~} 0 < x \le 1,\\ x & {\rm if~~} x \le 0 \end{cases} \]

its graph is:

Figure 4

  • From the logical / algorithmic point of view, one can say that a function of this kind has the peculiarity of being built using not only mathematical functions but also the logical function “if … then”.

3. The absolute-value function

  • In the chapter “Factorials, binomial coefficients and triangle inequality” of Part 2 the absolute value of a real number \(a\) was defined by

    \[ |a| = \begin{cases} a & {\rm if~~} a \ge 0\\ -a & {\rm if~~} a < 0 \end{cases} \]

    Associating with every \(x \in \mathbb{R}\) the number \(|x|\) we obtain a function of a real variable, which is the most important example of a piecewise defined function.

Definition 5: of absolute-value function

The absolute-value function is:

\[\begin{equation} \label{val_ass_f} f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto |x| \end{equation}\]

  • Sign. The absolute-value function is nonnegative in \(\mathbb{R}\) and positive in \(\mathbb{R} \setminus \{0\}\):

    \[ |x| \ge 0,~~ \forall x \in \mathbb{R} \qquad {\rm ~~and~~} \qquad |x| = 0 ~~\Longleftrightarrow~~ x=0. \]

    Indeed, if \(x \ge 0\) then \(|x|=x \ge 0\), and if \(x<0\) then \(|x|=-x>0\); in both cases \(|x|=0\) forces \(x=0\).

  • Monotonicity. The absolute-value function is decreasing in \((-\infty,0]\) and increasing in \([0,+\infty)\), but it is not monotonic in \(\mathbb{R}\).

    • If \(x_1 < x_2 \le 0\), then \(|x_1| = -x_1 > -x_2 = |x_2|\).

    • If \(0 \le x_1 < x_2\), then \(|x_1| = x_1 < x_2 = |x_2|\).

    • It is not monotonic in \(\mathbb{R}\) because \(f(-1)=1 > f(0)=0\) (hence it is not non-decreasing in \(\mathbb{R}\)) and \(f(0)=0 < f(1)=1\) (hence it is not non-increasing in \(\mathbb{R}\)).

  • Boundedness and image. The function is bounded below in \(\mathbb{R}\) (the number \(0\) is a lower bound for its values), but it is not bounded above: for every \(M \in \R\) with \(M>0\) we have \(f(M+1)=M+1>M\). Its image is

    \[ f(\mathbb{R}) = [0,+\infty), \]

    because \(|x| \ge 0\) for every \(x\) and, conversely, every \(y \ge 0\) is the image of \(y\) itself: \(|y|=y\).

  • Symmetry. The absolute-value function is even, since \(|-x|=|x|\) for every \(x \in \mathbb{R}\): its graph is symmetric with respect to the \(y\)-axis.

  • The graph consists of the two half-lines \(y=-x\) for \(x \le 0\) and \(y=x\) for \(x \ge 0\):

Figure 5