In high school, we learned the concepts of monotonicity and monotonic intervals of functions , which are consistent with what is taught in college calculus courses.

　　1. Monotone function:：monotonic increasing function and monotonic decreasing function

　　Definition: Let the domain of function be and the interval . If for any two points and on the interval , when , there is always

,

then the function is said to be a monotonically increasing function on the interval ; if for any two points and on the interval , when , there is always

,

then the function is said to be a monotonically decreasing function on the interval .

　　Monotonically increasing functions and monotonically decreasing functions are collectively called monotonic functions.

　　It is easy to see from the definition that the graph of a monotonically increasing function gradually rises along the positive direction of axis ( Figure 1), and the graph of a monotonically decreasing function gradually decreases along the positive direction of axis (Figure 2) .

Figure 1： [ Animation ]

Figure 2： [ Animation ]

　　For example, the parabola increases monotonically within x and decreases within , but it is not a monotonic function within (see Figure 3) . And is a monotonically increasing function within   (see Figure 4 ).

Figure 3： [ Animation ]　 Figure 4： [ Animation ]

　　2. Monotonicity and monotonic intervals of functions

　　If the function is a monotonic increasing function or a monotonic decreasing function on the interval , then the function is said to have (strict) monotonicity in this interval , and the interval is called the monotonic interval of the function .

　　Note: The monotonicity of a function is for a certain sub-interval within its domain of definition. The monotonicity of a function on a certain sub-interval reflects the changing trend of the function value on the interval and is the overall property of the function on the interval .