On a decomposition of \(C_ 0({\mathbb{R}}^ n)\) functions into simple component pieces (Q1067126)

The author shows that if f is a continuous real-valued function on \(R^ n\) and tends to zero at \(\infty\), then \(f=\sum_{k}\epsilon_ kb_ k\), where \(\epsilon_ k=\pm 1\) and each \(b_ k\) is continuous, non- negative, tends to zero at infinity and for each \(y\{b_ k>y\}\) is connected or else empty. Moreover, if \(b_ kb_{\ell}\not\equiv 0\), then \(\epsilon_ k=\epsilon_{\ell}\), and one of \(b_ k,b_{\ell}\) is positive and constant on the support of the other. The author also deals with a similar problem for Lipschitz functions.