Question

若f(x)的定义域是R,且f(x)=f(x-λ)+f(x+λ),则f(x)是周期函数且周期为6π

Answer 1

f(x)=f(x-λ)+f(x+λ)

=f(x-λ)+f(x)+f(x+2λ)

=f(x-λ)+f(x)+f(x+3λ)-f(x+4λ)

=f(x-λ)+f(x)+f(x+3λ)-(f(x+3λ)+f(x+5λ))

=f(x-λ)+f(x)-f(x+5λ)

所以f(x-λ)=f(x+5λ)

周期6λ