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
.
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}](img261.png) |
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
![$ [0, \pi /2 ]$](img267.png)
.
Take
and
in
. Then we have:
Example 2.5.5
Let
![$ g(x)=[x]+x^2$](img269.png)
.
Noah Dana-Picard
2007-12-28