A Lipschitz continuous function $\alpha:[0,a)\to[0,\infty)$, for $a>0$ is said to belong to class $\mathcal{K}$ if: 1. It is strictly increasing and; 2. $\alpha(0)=0$.

A Lipschitz continuous function $\alpha:[-b,a)\to[-\infty,\infty)$ is said to belong to extended class $\mathcal{K}$ if: 3. It is strictly increasing and; 4. $\alpha(0)=0$.