Problem
ALG-B3-M10-P007 Continuous Derivation Form
#7
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be continuous, additive, and satisfy \(f(xy)=xf(y)+yf(x)\). Find \(f\).
Continuous additivity gives \(f(x)=cx\).
Let \(f(x)=cx\). Then \(cxy=2cxy\) for all \(x,y\), so \(c=0\). The answer is \(f\equiv0\).
On \(\mathbb R\), continuity plays the role of rationality.