Monotonous functions.

Definition 2.5.2   Let denote a function defined on an interval .
1. The function is strictly increasing on if .
2. The function is increasing on if .
3. The function is strictly decreasing on if .
4. The function is decreasing on if .
5. The function is (strictly) monotonous on if it is either (strictly) increasing on or (strictly) decreasing on .
6. The function is constant on if .

For example, an affine function is strictly increasing on when , strictly decreasing on when and constant on when .

Be careful! A function can increase on an interval and decrease on another interval. For example the function (the graph is displayed on Figure 5(a)):

• decreases on ;
• increases on ;
• is not monotonous on .

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

• increases on and on ;
• decreases on .
• is not monotonous on .

Example 2.5.3
(i)
The absolute value function, whose graph is displayed in Figure fig abs value is strictly decreasing over and strictly increasing over . This function is not monotonous over .
(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 .

Example 2.5.4   Let . We show that the function increases strictly over the interval .

Take and in . Then we have:

Example 2.5.5   Let .

Noah Dana-Picard 2007-12-28