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It was claimed in [Sadeqi, I.—Salehi Paydar, M.:  A comparative study of Ky Fan hemicontinuity and Brezis pseudomonotonicity of mappings and existence results , J. Optim. Theory Appl. 165(2) (2015), 344–358] that the gradient of a convex Gâteaux differentiable function is Fanhemicontinuous. The aim of the present paper is to correct this implication by exemplifying for∇  ‖  ⋅  ‖$\nabla \|\cdot \|$   in a Hilbert space. This goal was achieved by considering its domain to be scalarly-positive (see Definition 3). For a perturbation of a Fan-hemicontinuous operator with∇  ‖  ⋅  ‖  ,$\nabla \|\cdot \|,$   weak compactness for the set of solutions of the variational inequality problem is obtained.

Fan-Hemicontinuity for the gradient of the norm in Hilbert space