Monotonous functions.

Definition 2.5.2   Let $f$ denote a function defined on an interval $I$ .

1. The function $f$ is strictly increasing on $I$ if $\forall x_1 \in I, \forall x_2 \in I, \; x_1 < x_2 \Longrightarrow f(x_1) < f(x_2)$ .
2. The function $f$ is increasing on $I$ if $\forall x_1 \in I, \forall x_2 \in I, \; x_1 < x_2 \Longrightarrow f(x_1) \leq f(x_2)$ .
3. The function $f$ is strictly decreasing on $I$ if $\forall x_1 \in I, \forall x_2 \in I, \; x_1 < x_2 \Longrightarrow f(x_1) > f(x_2)$ .
4. The function $f$ is decreasing on $I$ if $\forall x_1 \in I, \forall x_2 \in I, \; x_1 < x_2 \Longrightarrow f(x_1) \leq \geq f(x_2)$ .
5. The function $f$ is (strictly) monotonous on $I$ if it is either (strictly) increasing on $I$ or (strictly) decreasing on $I$ .
6. The function $f$ is constant on $I$ if $\forall x_1 \in I, \forall x_2 \in I, \; f(x_1)=f(x_2)$ .

For example, an affine function $f: x \mapsto ax+b$ is strictly increasing on $\mathbb{R}$ when $a>0$ , strictly decreasing on $\mathbb{R}$ when $a<0$ and constant on $\mathbb{R}$ when $a=0$ .

Be careful! A function can increase on an interval and decrease on another interval. For example the function $f: x \mapsto x^2$ (the graph is displayed on Figure 5(a)):

• decreases on $(-\infty, 0)$ ;
• increases on $(0,+\infty)$ ;
• is not monotonous on $\mathbb{R}$ .

Figure 5: Functions which are monotonous by intervals.

The function $g$ whose graph is displayed on Figure 5(b):

• increases on $(0,1)$ and on $(4,9/4)$ ;
• decreases on $(1,4)$ .
• is not monotonous on $(0,9/4)$ .

Example 2.5.3       

(i)

The absolute value function, whose graph is displayed in Figure fig abs value is strictly decreasing over $(-\infty , 0]$ and strictly increasing over $[0,+\infty )$ . This function is not monotonous over $\mathbb{R}$ .

(ii)

The floor function (integer part function) whose graph is displayed in Figure fig integer part, is an increasing function, but not a strictly increasing function over $\mathbb{R}$ .

Example 2.5.4   Let $f(x)=2x+ \sin x -1$ . We show that the function $f$ increases strictly over the interval $[0, \pi /2 ]$ .

Take $x_1$ and $x_2$ in $[0, \pi /2 ]$ . Then we have:

$$x_1 < x_2 \Longrightarrow \begin{cases}2x_1 < 2x_2 \sin x_1 < ... ...Longrightarrow 2x_1+ \sin x_1 < 2x_2+ \sin x_2 \Longrightarrow f(x_1) < f(x_2).$$

Example 2.5.5   Let $g(x)=[x]+x^2$ .