Ioffe, A.; Miłosz, T. On a characterization of \(C^{1,1}\) functions. (English) Zbl 1029.49017 Cybern. Syst. Anal. 38, No. 3, 313-322 (2002) and Kibern. Sist. Anal. 2002, No. 3, 3-13 (2002). Let \(f\) be a real-valued locally Lipschitz function on the Banach space \(X\). After a dicussion of the basic properties of the generalized second-order derivative \[ f^{\circ\circ}(x; h,e)= \limsup_{u\to x; t,\tau\to+0} {f(u+ th+\tau e)- f(x+ th)- f(x+ \tau e)+ f(x)\over t\tau} \] it is shown that the following conditions are equivalent:(1) \(f\) is \(C^{1,1}\), i.e., continuously Fréchet differentiable and its derivative (as vector-valued mapping) is locally Lipschitz;(2) \(f\) is Clarke regular and the Clarke directional derivative \(f^\circ(\cdot,\cdot)\) is locally Lipschitz;(3) the Dini-Hadamard subdifferential \(\partial^-f(x)\) is nonempty for all \(x\) and (as set-valued mapping) locally Lipschitz;(4) the lower Dini-Hadamard directional derivative \(d^-f(\cdot,\cdot)\) is locally Lipschitz;(5) \(f\) is locally second-order Lipschitz, i.e., \(f^{\circ\circ}(\cdot,\cdot,\cdot)\) is locally bounded. Reviewer: Jörg Thierfelder (Ilmenau) Cited in 1 Document MSC: 49J52 Nonsmooth analysis Keywords:Lipschitz condition; directional derivative; Clarke regularity; semicontinuity; convexity; Clarke subdifferential; Lipschitz function; generalized second-order derivative; Dini-Hadamard subdifferential; Dini-Hadamard directional derivative PDFBibTeX XMLCite \textit{A. Ioffe} and \textit{T. Miłosz}, Cybern. Syst. Anal. 38, No. 3, 313--322 (2002) and from Kibern. Sist. Anal. 2002, No. 3, 3--13 (2002; Zbl 1029.49017) Full Text: DOI