Problem
ALG-B3-M03-P017 Affine function from a lower bound
#17
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\), \(f(x+y)=f(x)+f(y)-3\), and \(f(x)\ge -5\) for all \(x\in[0,1]\). Prove that \(f(x)=cx+3\) for some \(c\).
Hint. First shift the function, then obtain boundedness of the additive part.
Let \(g(x)=f(x)-3\). Then \(g\) is additive. On \([0,1]\), \(g(x)\ge -8\). Also \(g(x)=g(1)-g(1-x)\le g(1)+8\), so \(g\) is bounded on \([0,1]\). Therefore \(g(x)=cx\), and \(f(x)=cx+3\).
Goal: show which conditions actually force a function to be linear or affine.