A functional equation for the cotangent on the open unit interval (Q5931971)

The author considers the functional equation \[ f(x)=\tfrac{1}{2} f(\tfrac{x}{2})+\tfrac{1}{2} f(\tfrac{x+1}{2}), \quad 0<x<1 \tag{*} \] and, after having noticed that any function defined on \([1/4,3/4]\) satisfying the condition \[ 2f(\tfrac{1}{2})=f(\tfrac{1}{4})+f(\tfrac{3}{4}) \] may be extended to a solution of (*), proves a representation theorem for the solutions of (*) which are Riemann integrable on \([1/4,3/4]\). As a corollary, we have that if a solution \(f\) of (*), Riemann integrable on \([1/4,3/4]\) is such that the limits \[ \lim_{x \to 0^+} xf(x)=a \qquad \text{and} \qquad \lim_{x \to 0^+} xf(1-x)=b \] both exist, then \(b=-a\) and \[ f(x)=a\pi \cot \pi x+f(\tfrac{1}{2}). \]