Problem
ALG-B3-M11-P017 Parametric Shift
#17
★★★★★ Level 5 of 5
Find all pairs \((a,f)\), where \(f\) is continuous, \(f(x+y)=f(x)+f(y)+a xy\), and \(f(f(x))=x+a\).
First the general form, then degree of composition.
The general form is \(f(x)=\frac a2x^2+cx\). If \(a\ne0\), then \(f(f(x))\) has degree \(4\), impossible. Hence \(a=0\). Then \(f(x)=cx\), and \(c^2x=x\), so \(c=1\) or \(c=-1\). The answer is \(a=0\), \(f(x)=x\), \(f(x)=-x\).
Parametric mixed problem.