# Lipschitz Functions with Prescribed Derivatives and Subderivatives

Borwein, Jonathan M. and Moors, Warren B. and Wang, Shawn Xianfu (1997) Lipschitz Functions with Prescribed Derivatives and Subderivatives. Journal of Nonlinear Analysis: Theory, Methods & Applications, 29 (1). pp. 53-63. ISSN 0362546X

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Postscript
In general it is difficult to construct Lipschitz functions which are not directly built up from either convex or distance functions. One impediment to such constructions is that outside of the real line it is difficult to find anti-derivatives. The main result of this paper provides, under suitable circumstances, a technique for constructing such anti-derivatives. More precisely, we show that if $f_{1}$, $f_{2}$,$\dots$, $f_{n}$ are continuously differentiable real-valued locally Lipschitz functions defined on a non-empty open subset $A$ of a separable Banach space $X$, then there exists a real-valued locally Lipschitz function $g$ defined on $A$ such that at each point $x\in A$ the Clarke subgradient of $g$ at $x$ equals $co\{\nabla f_{1}(x), \nabla f_{2}(x),\dots, \nabla f_{n}(x)\}$. This same construction also shows that for any finite family $\{T_{1},T_{2},\dots, T_{n}\}$ of maximal cyclically monotone mappings from $A$ into non-empty subsets of $X^{*}$, there exists a real-valued locally Lipschitz function $g$ defined on $A$ such that at each point $x\in A$ the Clarke subgradient of $g$ at $x$ equals $co\{ T_{1}(x), T_{2}(x),\dots, T_{n}(x)\}$. Moreover, we show that $g$ is convex if and only if $T_{1}=T_{2}=\cdots=T_{n}$.