Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ Jonathan M. Borwein Xianfu Wang 13 February 2006 2006 2 13 491 496 723-5250-BoWa-2005 10.wxyz/C2005V72P3p491 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5250-BoWa/ In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire's category) have a maximal Clarke subdifferential. In the present paper, we show that in a separable Banach space the set of non-expansive Lipschitz functions with a maximal Clarke subdifferential is not only of generic, but also staunch. 49J52 Research for the first author was supported by NSERC and the CRC programme. Research for the second author was supported by NSERC. J.M. Borwein and X. Wang Proc. Amer. Math. Soc. MR1777577 J.M. Borwein and X. Wang; Lipschitz functions with maximal subdifferentials are generic, \textit{Proc. Amer. Math. Soc.} \textbf{128} (2000), pp.~3221--3229. J.M. Borwein, W.B. Moors and X. Wang Trans. Amer. Math. Soc. MR1837212 J.M. Borwein, W.B. Moors and X. Wang; Generalized subdifferentials: a Baire categorical approach, \textit{Trans. Amer. Math. Soc.} \textbf{353} (2001), pp.~3875--3893. F.H. Clarke Wiley Interscience MR709590 F.H. Clarke; \textit{Optimization and nonsmooth analysis} (Wiley Interscience, New York, 1983). J.R. Giles and S. Sciffer Bull. Austral. Math. Soc. MR1210135 J.R. Giles and S. Sciffer; Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces, \textit{Bull. Austral. Math. Soc.} \textbf{47} (1993), pp.~205--212. S. Reich, A.J. Zaslavski C. R. Acad. Sci. Paris MR1860926 S. Reich, A.J. Zaslavski; The set of noncontractive mappings is $\sigma $-porous in the space of all non-expansive mappings, \textit{C. R. Acad. Sci. Paris} \textbf{333} (2001), pp.~539--544. L. Zajicek Colloq. Math. MR1628994 L. Zajicek; Small non-$\sigma $-porous sets in topologically complete metric spaces, \textit{Colloq. Math.} \textbf{77} (1998), pp.~293--304.