Other Properties - Describing Functions

Mathematical Writing - Vivaldi Franco 2014

Other Properties
Describing Functions

A function of the form $$f(x)=ax$$, where $$a$$ is a real number, is said to be linear. The term linear originates from ’line’, and functions of the type $$f(x)=ax+b$$, are sometimes referred to as being ’linear’ because their graph is a line (the correct term is affine). This acceptation of the term linear is common for polynomial functions, which in the first instance are characterised by their degree. So we speak of a linear, quadratic, cubic, quartic function, etc.

Fig. 5.3

A continuous piecewise affine function

Fig. 5.4

Left A step function. (The vertical segments are just a guide to the eye; they are not part of the graph of the function.) Right A periodic continuous function which is piecewise differentiable

Consider the absolute value function, defined as follows:

$$\begin{aligned} |x|={\left\{ \begin{array}{ll}\,x&{} \mathrm{{if}}\,x\geqslant 0\\ -x&{} \mathrm{{if}} x < 0.\\ \end{array}\right. } \end{aligned}$$

(5.13)

This function is not linear, but it is made of two linear pieces, glued together at the origin. Functions made of linear or affine pieces are said to be piecewise linear or piecewise affine [see Eq. (5.3) and Fig. 5.3]. A function is piecewise defined if its domain is partitioned into disjoint intervals or rays, with the function being specified independently over each interval. The properties of a piecewise-defined function may fail at the end-points of the intervals of definition. In this case terms such as piecewise increasing, piecewise continuous, piecewise differentiable, piecewise smooth may be used—see Fig. 5.4. A piecewise constant function is called a step-function.

EXAMPLE. The floor of a real number $$x$$, denoted by $$\lfloor x\rfloor $$, is the largest integer not exceeding $$x$$. Similarly, the ceiling $$\lceil x\rceil $$ represents the smallest integer not smaller than $$x$$. Floor and ceiling are the prototype step functions. Closely connected to the floor is the fractional part of a real number $$x$$, denoted by $$\{x\}$$. (The notational clash with the set having $$x$$ as its only element is one of the most spectacular in mathematics!) The fractional part is defined as

$$ \{x\}:= x-\lfloor x\rfloor $$

from which it follows that $$0\leqslant \{x\}<1$$. This function is discontinuous and piecewise affine.

EXAMPLE. Describe the following function: [ $$/\!\!\varepsilon $$ ]

This is a smooth function, which is bounded and non-negative. It features an infinite sequence of evenly spaced local maxima, whose height decreases monotonically to zero for large arguments. There is one zero of the function between any two consecutive local maxima.

We rewrite it, borrowing some terminology from physics.

This function displays regular oscillations of constant period, with amplitude decreasing monotonically to zero.

The following functions have a behaviour that is qualitatively similar to that displayed above.

$$ f(x)=\sin ^2(x)\mathrm{e}^{-x} \qquad \,\, f(x)=\frac{1-\cos (x)}{1+x^2}. $$