
Modifications of Whitney's $C^1$ extension theorem.
Dovhoruk, Olesya ; Zajíček, Luděk (advisor) ; Holický, Petr (referee)
Title: Modifications of Whitney's C1 extension theorem. Author: Olesya Dovhoruk Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Luděk Zajíček, DrSc., Department of Mathematical Ana lysis Abstract: This work deals with modifications of the Whitney's C1 extension theorem on a special closed set M in Rn . The work investigates of whether it is possible to skip some of the assumptions of the Whitney's theorem. It turns out that if we do not assume the continuity of a function f : M → R, which is being extended from a general closed set M ⊂ Rn , then f is continuous from the remaining assumptions in the Whitney's theorem, but if we skip the continuity of a function d, which features in the Whitney's theorem (and plays a role of the generalised differential of the function f), then d is continuous from the remaining assumptions, but just for n = 1. Further, some proposals based on modifications of the assumptions of the Whit ney's theorem are proved. For instance, the theorem of an existence of a C1 extension of a function f : (a, b)×[0, c), f ∈ C1 ((a, b)×(0, c)), for which is valid that the function gradf has finite limits in (a, b) × {0}. Another similar result of the work is the existence of a C1 extension of a function f : M ⊂ Rn → R, where M = M◦ = ∅ is a compact and convex set and...
