Function is a hot spot in college entrance examination every year, and the application of abstract function properties is one of the difficulties of function. Abstract function refers to some properties or algorithms that a function satisfies without giving a specific analytic function or image. This kind of function test questions can not only comprehensively examine students' understanding of function concepts and algebraic reasoning and argumentation ability, but also comprehensively examine students' understanding and acceptance of mathematical symbol language and their understanding of general and special relations. Therefore, it is favored by proposers and appears constantly in college entrance examination questions in recent years. However, due to the abstract nature and hidden nature of such problems, most students feel helpless when solving such problems. Here are some examples to discuss the solutions to these problems.

Example: Let y=f(x) be a function defined in the interval [- 1, 1] and satisfy the following conditions:

(I)f(- 1)= f( 1)= 0；

(ii) For any u, v ∈ [- 1, 1], there is -f (u)-f (v)-≤-u-v-.

(i) Prove that for any x ∈ [- 1, 1], there exists x-1≤ f (x) ≤1-x;

(2) Prove that for any u, v ∈ [- 1], there exists -f (u)-f (v)-≤ 1.

Solve the problem:

(1) Proof: According to the title conditions, when x ∈ [- 1], there is f (x) = f (x)-f (1) ≤-x-1-= 65433.

(2) Prove that for any u, v ∈ [- 1], when -u-v-≤ 1, there is -f (u)-f (v)-≤ 1.

When-u-v->; 1, u v<0, you might as well set u; 1, where v ∈ (0, 1) and u ∈ [- 1, 0).

In order to make the known conditions work, a point on [- 1, 0] must be matched with u to make use of the known conditions. Combining f (- 1) = f (1) = 0, this point can be selected as-1. Similarly, the point 1 on (0, 1) should be taken to match v to take advantage of the known conditions. Therefore,-f (u)-f (v)-≤-f (u)-f (-1)-+-f (v)-f (1)-≤-u+1-+-v-6544. 1

To sum up, any u, v ∈ [- 1, 1] has -f (u)-f (v)-≤ 1.

Comments: Regarding the problem of abstract function, the equality or inequality that the function satisfies is often given. Therefore, when solving related problems, we should first change the structure of the formula to be proved or solved, so that the structure of the problem to be proved or solved is the same as that of the known problem. For example, in this question, the analytic expression of the function y=f(x) is not given, but a group of specific corresponding relations f(- 1)=f( 1)=0, and the general relationship that the absolute value of the difference between two variables is not less than the absolute value of the difference between corresponding function values is given. In the proof of (1), f(x) is rewritten as-f (x)-=-f (x)-f (1)-; In the proof of (2), f (- 1) = f (1) = 0 and -f (u)-f (v)-f (u)-≤-f (-).

In addition, the function properties given in the abstract function problem often hold true for all real numbers in the domain. Therefore, according to the meaning of the question, it is one of the most important strategies to specialize the general problem and select appropriate special values (such as x= 1, Y = 0, etc.). ).

In short, it is generally difficult to solve abstract function problems by conventional methods, but if we can solve them by special methods and means through information analysis and research on the topic, we will often get twice the result with half the effort, and at the same time, we should closely cooperate with each other and complement each other in the use of these strategies.