For a multivalued map ϕ: Y ⊸ (X, τ ) between topological spaces, the upper semifinite topology A(τ ) on the power set A(X) = {A ⊂ X : A ≠ ∅} is such that ϕ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map ϕ: Y → (A(X), A(τ )). In this paper, we seek a result like this from a reverse viewpoint, namely, given a set X and a topology Γ on A(X), we consider a natural topology R(Γ) on X, constructed from Γ satisfying R(Γ) = τ if Γ = A(τ ), and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map ϕ: Y ⊸ (X, R(Γ)) to be equivalent to the continuity of the singlevalued map ϕ: Y → (A(X), Γ).
For a differentiable function f : I → ℝ k , where I is a real interval and k ∈ ℕ, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean M : I 2 → I such that f (x) − f (y) = (x − y)f ′ (M(x,y)), x,y ∈ I, are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.