We establish a mixed convex-Lipschitz mean value inequality from which recent results of Clarke and Ledyaev and of Lewis and Ralph follow naturally. We also provide various refinements and extensions. Finally, we answer affirmatively several open questions on the existence of ``squeeze'' theorems for a finite number of Lipschitz functions.

We study the conjugate of the maximum, f ∨ g f \vee g , of f f and g g when f f and g g are proper convex lower semicontinuous functions on a Banach space E E . We show that ( f ∨ g ) ∗ ∗ = f ∗ ∗ ∨ g ∗ ∗ (f \vee g)^{**} = f^{**} \vee g^{**} on the bidual, E ∗ ∗ E^{**} , of E E provided that f f and g g satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for ( f ∨ g ) ∗ (f \vee g)^{*} and for the “preconjugate” of f ∗ ∨ g ∗ f^{*}\vee g^{*} .

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.

The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.

P.S. Kenderov has shown that every monotone operator on an Asplund Banach space is continuous on a dense Gδ subset of the interior of its domain. We prove a general result which yields as special cases both Kenderov's Theorem and a theorem of Collier on the Fréchet differentiability of weak* lower semicontinuous convex functions.