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On a globalization property

Stefan Rolewicz (1993)

Applicationes Mathematicae

Let (X,τ) be a topological space. Let Φ be a class of real-valued functions defined on X. A function ϕ ∈ Φ is called a local Φ-subgradient of a function f:X → ℝ at a point x 0 if there is a neighbourhood U of x 0 such that f(x) - f( x 0 ) ≥ ϕ(x) - ϕ( x 0 ) for all x ∈ U. A function ϕ ∈ Φ is called a global Φ-subgradient of f at x 0 if the inequality holds for all x ∈ X. The following properties of the class Φ are investigated: (a) when the existence of a local Φ-subgradient of a function f at each point implies...

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