- 设f(x)是定义在R是的函数,证明|f(x)|=f(x)sgn[?
- 设f(x)是定义在R是的,证明|f(x)|=f(x)sgn[f(x)]
y=sgn(x)是一个函数,
x=0,y=0
x>0,y=1
x<0,y=-1
请高手证明!谢.
- 设f(x)是定义在R是的函数,证明|f(x)|=f(x)sgn[f(x)]
y=sgn(x)是符号函数
x=0,y=0
x>0,y=1
x<0,y=-1
由符号函数sgn(x)的定义：
f(x)＞0时:sgx[f(x)]=1--->|f(x)|=f(x)=f(x)*1=f(x)sgn[f(x)]
f(x) =0时:sgx[f(x)]=0--->|f(x)|=0=f(x)sgn[f(x)]
f(x)＜0时:sgx[f(x)]=-1--->|f(x)|=-f(x)=f(x)(-1)=f(x)sgn[f(x)]
综上：|f(x)|=f(x)sgn[f(x)]