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Recently the notions of monotone, demicontinuous and hemicontinuous functions have been introduced in connection with nonlinear problems in functional analysis (Browder [ l ; 2; 3; 4; 5], Minty [6; 7; 8]). The object of the present note is to show that under rather general conditions, hemicontinuity is equivalent to demicontinuity for monotone functions. Let X be a (real or complex) Banach space and X* its adjoint space as the set of all bounded conjugate-linear functionals on X. The value of f (EX* a t uÇiX is denoted by (ƒ, u). We use the notations —> and —̂ for strong convergence in X (or in X* or in the set of real numbers) and weak* convergence in X*, respectively. Let G be a function from X to X* with domain D~D(G) C.X. G is said to be demicontinuous if un(E.D, n — 1, 2, 3, • • -, uÇ^D and H/ji 'U imply Gun-^Gu. G is hemicontinuous if uÇzD, vÇîX and u+tnv£D, where tn is a sequence of positive numbers such that tn-*0, imply G(u+tnv)—*Gu. We shall say that G is locally bounded if unÇ.D} uÇ^D and un—*u imply that Gun is bounded. Obviously a demicontinuous function is hemicontinuous and locally bounded. G is said to be monotone if Re(Gu — Gv, u — v) ^ 0 for w, VÇE.D. These definitions may be void if D is too arbitrary. In what follows we shall assume that D is quasi-dense. By this we mean that for each uÇiD there is a dense subset Mu of X such that for each vCzMu, u-\-tvÇzD for sufficiently small / > 0 (the smallness of / depending on v). Thus any open subset of X as well as any dense linear manifold of X is quasi-dense.

Demicontinuity, hemicontinuity and monotonicity