- The function is strictly increasing on if .
- The function is increasing on if .
- The function is strictly decreasing on if .
- The function is decreasing on if .
- The function is (strictly) monotonous on if it is either (strictly) increasing on or (strictly) decreasing on .
- 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 .

- (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 .

Take and in . Then we have:

Noah Dana-Picard 2007-12-28