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functional equation f(x+y)=f(x)f(a-y)+f(y)f(a-x) - show f is constant

Source: bmo 1987

April 23, 2007

function

Problem Statement

Let

a

be a real number and let

f : \mathbb{R}\rightarrow \mathbb{R}

be a function satisfying

f(0)=\frac{1}{2}

and f(x+y)=f(x)f(a-y)+f(y)f(a-x), \forall x,y \in \mathbb{R}. Prove that

f

is constant.