On a globalization property

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 ${\displaystyle x_0}$ if there is a neighbourhood U of ${\displaystyle x_0}$ such that f(x) - f(${\displaystyle x_0}$) ≥ ϕ(x) - ϕ(${\displaystyle x_0}$) for all x ∈ U. A function ϕ ∈ Φ is called a global Φ-subgradient of f at ${\displaystyle x_0}$ 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 the existence of a global Φ-subgradient of f at each point (globalization property), (b) when each local Φ-subgradient can be extended to a global Φ-subgradient (strong globalization property).