Resolution of an integral equation with the Thue-Morse sequence (Q441351)

The author shows that there exists a continuous function \(f(x)\) valued in \([-1,1]\) solving the equation NEWLINE\[NEWLINE\int^x_0 f(t)\,dt+ f(0)= f\Biggl({x\over 2}\Biggr)NEWLINE\]NEWLINE and having the following properties:NEWLINENEWLINE(1) for each integer \(n\), \(f(2n+1)= u_n\) and \(f(2n)= 0\),NEWLINENEWLINE (2) for each negative real number \(x\), \(f(x)= 0\),NEWLINENEWLINE (3) for each positive real number \(x\), \(|f(x)|= |f(x+2)|\), where \(u_n\) is the Thue-Morse sequence.