Point derivations for Lipschitz functions andClarke's generalized derivative

Clarke's generalized derivative ${\displaystyle f^0(x,v)}$ is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed ${\displaystyle x\in X}$ and fixed ${\displaystyle v\in E}$ the function ${\displaystyle f^0(x,v)}$ is continuous and sublinear in ${\displaystyle f\in Lip(X,d)}$. It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz's product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.