Norm { to { Weak Upper Semicontinuous Monotoneoperators

In any Banach space a monotone operator with a norm-to-weak upper semicontinuous multivalued selection on an open set D is singlevalued and norm-to-norm upper semicontinuous at the points of a dense G subset of D. Monotone operators | and especially a special case of them, subdiierentials of convex functions | play an important role in various parts of nonlinear analysis. One of the often investigated problems is the question about generic continuity of monotone operators, which in the case of subdiierentials means generic Fr echet diierentiability of convex functions. In general Banach spaces monotone operators are not always generically continuous. There are numerous characterizations of Asplund spaces, i.e. the spaces in which any monotone operators is generically continuous on the interior of its eeective domain (see e.g. Ph]). The aim of this note is to prove that monotone operator of a certain class are generically continuous in an arbitrary Banach space (Theorem 2). Our result can be deduced from a selection result by Ch. Stegall St], based on hard topological techniques (Remark 2). We present a relatively simple alternative proof, self-contained in the sense that it uses monotone operator techniques only. Three main tools of this note are following: a slight modiication of a result by D. Preiss and L. Zaj cek Pr{Zaj] (Theorem P{Z), the observation that the image of a separable set by a norm-to-weak continuous mapping is separable, and the well known method of separable reduction which extends the result from the separable into the nonseparable case. Definitions and Notations Let us recall some deenitions and notations. X always denotes a real Banach space, X its continuous dual. Closed and open balls with centre c (in X or X) and radius r > 0 are denoted by B(c; r) and B(c; r), respectively.