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Integer part and fractional part functions

Part 2 · Functions · Chapter 7 · lecture notes by Fabio Furini · Chapter 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 \(f(x)= \lfloor x \rfloor\) is monotone increasing, like the upper integer part function \(f(x)= \lceil x \rceil\).

  • 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”.