Problem
ALG-B3-M09-P006 Bound on a Symmetric Interval
#6
★★★☆☆ Level 3 of 5
Let \(f\) be additive, \(|f(x)|\le7\) for \(|x|\le1\), and \(f(2)=10\). Find \(f\).
Boundedness gives \(f(x)=cx\).
Boundedness on an interval implies linearity: \(f(x)=cx\). Since \(f(2)=10\), \(2c=10\), so \(c=5\). The answer is \(f(x)=5x\).
Reinforces boundedness with a different normalization.