Problem
ALG-B3-M05-P007 An Irrational Input Value
#7
★★★☆☆ Level 3 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive and continuous, and suppose \(f(\sqrt{3})=6\). Find \(f(x)\).
A continuous additive function is linear.
Since \(f\) is continuous and additive, \(f(x)=cx\). From \(c\sqrt{3}=6\), we get \(c=\frac{6}{\sqrt{3}}=2\sqrt{3}\). The answer is \(f(x)=2\sqrt{3}x\).
Shows the role of regularity on \(\mathbb R\).