关于 是否存在这样的函数( 个人觉得是 常微分方程问题)Is there such a function?suppose g is a function satisfying the following two preperties:(a).g(x)=xg(1/x),for all real numbers x( x is not zero),and(b).g(x)+g(y)=1+g(x+y) f
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关于 是否存在这样的函数( 个人觉得是 常微分方程问题)Is there such a function?suppose g is a function satisfying the following two preperties:(a).g(x)=xg(1/x),for all real numbers x( x is not zero),and(b).g(x)+g(y)=1+g(x+y) f
关于 是否存在这样的函数( 个人觉得是 常微分方程问题)
Is there such a function?
suppose g is a function satisfying the following two preperties:
(a).g(x)=xg(1/x),for all real numbers x( x is not zero),and
(b).g(x)+g(y)=1+g(x+y) for any two real numbers x,y.
Determine all such functions or prove that there is no such function
关于 是否存在这样的函数( 个人觉得是 常微分方程问题)Is there such a function?suppose g is a function satisfying the following two preperties:(a).g(x)=xg(1/x),for all real numbers x( x is not zero),and(b).g(x)+g(y)=1+g(x+y) f
When x=0,
g(0)+g(0)=1+g(0)
so g(0)=1
Then
g(1/x)+g(-1/x)=2
g(x)=xg(1/x)
g(x)/x+g(-x)/-x=2
So,g(x)-g(-x)=2x
g(x)+g(-x)=2
Therefore,g(x)=x+1