Problem
ALG-B3-M05-P008 Boundedness on an Interval
#8
★★★☆☆ Level 3 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive, bounded on \([0,1]\), and \(f(1)=2\). Prove that \(f(x)=2x\).
An additive function bounded on an interval is continuous at zero.
By the standard fact, an additive function bounded on an interval is continuous at zero and hence everywhere. Therefore \(f(x)=cx\). Since \(f(1)=2\), we get \(c=2\). Thus \(f(x)=2x\).
Can be used as a theoretical support before harder problems.