- Title
- A chain rule for essentially smooth Lipschitz functions
- Creator
- Borwein, Jonathan M.; Moors, Warren B.
- Relation
- SIAM Journal on Optimization Vol. 8, Issue 2, p. 300-308
- Publisher Link
- http://dx.doi.org/10.1137/S1052623496297838
- Publisher
- Society for Industrial and Applied Mathematics (SIAM)
- Resource Type
- journal article
- Date
- 1998
- Description
- In this paper we introduce a new class of real-valued locally Lipschitz functions (that are similar in nature and definition to Valadier's saine functions), which we call arcwise essentially smooth, and we show that if g : Rm → R is arcwise essentially smooth on Rm and each function fj : R^n → R, 1 ≤ j ≤ m, is strictly differentiable almost everywhere in Rn, then g ○ f is strictly differentiable almost everywhere in Rn, where f ≡ (f₁,f₂,...,fm). We also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well behaved.
- Subject
- Lipschitz functions; chain rule; Haar-null sets; differentiability; essentially smooth
- Identifier
- http://hdl.handle.net/1959.13/940494
- Identifier
- uon:13023
- Identifier
- ISSN:1052-6234
- Language
- eng
- Full Text
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