问题:已知f(x)是定义在 R上的函数,且对于任意的 a,b∈R都满足 f(a·b)=af(b)+bf(a),求证:f(x)是常数函数.本题源自文[1]中第一章第一节的补充题.一位刚入大学校门的学生向我求助,经过仔细思考,我认为原题是错题.解答:f(x)=c·xln|x|,x≠0,0,x=0(c 是常数)是满足 Problem: It is known that f (x) is a function defined on R and f (a · b) = af (b) + bf (a) for any a and b∈R. Is a constant function .This question from the article [1] in the first chapter of the first quarter supplementary question.A student who just entered the university gate to me for help, after careful consideration, I think the original title is the wrong question.Answer: f ( x) = c · xln | x |, x ≠ 0,0, x = 0 (c is a constant) is satisfied