Abstract: We study monotone operators on quasi open or convex subsets of a real Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (quasi open means that the contingent cone at each point equals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Among others we characterize the maximality of such an operator in terms of its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>w</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{w^*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-upper semicontinuity properties and, in the case of a convex domain, also in terms of its behavior at the support points. We next give sufficient conditions for such an operator to be generically single valued, extending Kenderov’s theorems. As an application we reobtain generic Gâteaux and Fréchet differentiability results for convex functions defined on not necessarily open convex sets.