Problem
ALG-B3-M03-P020 Family or unique answer
#20
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\), \(f(x+y)=f(x)+f(y)-2\). Describe all solutions. Then add the condition that \(f\) is nondecreasing and find all solutions under the additional condition \(f(1)=5\).
Hint. First put \(A(x)=f(x)-2\).
Let \(A(x)=f(x)-2\). Then \(A(x+y)=A(x)+A(y)\). Without extra conditions, all solutions are \(f(x)=A(x)+2\), where \(A\) is arbitrary additive. If \(f\) is nondecreasing, then \(A\) is nondecreasing, so \(A(x)=cx\). The condition \(f(1)=5\) gives \(c+2=5\), hence \(c=3\). Thus \(f(x)=3x+2\).
Goal: show which conditions actually force a function to be linear or affine.