Problem
ALG-B3-M10-P001 Parameter \(a\)
#1
★★★★☆ Level 4 of 5
Find all continuous \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)=f(x)+f(y)+a xy\).
Subtract \(\frac a2x^2\).
Let \(g(x)=f(x)-\frac a2x^2\). Then \(g(x+y)=g(x)+g(y)\). By continuity, \(g(x)=cx\). The answer is \(f(x)=\frac a2x^2+cx\), where \(c\in\mathbb R\). Direct checking works.
Basic parametric advanced problem.