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)):

Figure 5: Functions which are monotonous by intervals.
\begin{figure}\mbox{
\subfigure[]{\epsfig{file=parabola1.eps,height=4cm}}
\qquad \qquad
\subfigure[]{\epsfig{file=BrokenLine1.eps,height=4cm}}
}\end{figure}

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

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:

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

Example 2.5.5   Let $ g(x)=[x]+x^2$ .

Noah Dana-Picard 2007-12-28