Problem
ALG-B3-M10-P013 Parameter and Shift
#13
★★★★★ Level 5 of 5
Find all pairs \((a,f)\), where \(f\) is continuous, \(f(x+y)=f(x)+f(y)+a xy\), and \(f(f(x))=x+1\).
Again first find the general form.
As before, \(f(x)=\frac a2x^2+cx\). If \(a\ne0\), the composition has degree \(4\), impossible. Thus \(a=0\), \(f(x)=cx\). Then \(f(f(x))=c^2x\), which cannot equal \(x+1\). Therefore there are no solutions.
Variation with an impossible shift.