TABLE DES MATIÈRES Introduction.................................................................................................................................................................. 5 Chapitre I. Conditions équivalentes à la mesurabilité d'une fonction de deux variables..................................... 7 Chapitre II. Conditions suffisantes pour la mesurabilité des fonctions de deux variables................................. 18 Chapitre III. Mesurabilité des fonctions de deux variables dont les coupes sont des dérivées......................... 25 Chapitre IV. Caractérisation des fonctions ayant la propriété (G).............................................................................. 33 Chapitre V. Mesurabilité de la superposition F(x, f(x)).................................................................................................. 35 Problèmes ouverts.............................................................................................................................................................. 43 Travaux cités......................................................................................................................................................................... 44
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
A function f: ℝⁿ → ℝ satisfies the condition $Q_{i}(x)$ (resp. $Q_{s}(x)$, $Q_{o}(x)$) at a point x if for each real r > 0 and for each set U ∋ x open in the Euclidean topology of ℝⁿ (resp. strong density topology, ordinary density topology) there is an open set I such that I ∩ U ≠ ∅ and $|(1/μ (U∩I)) ∫_{U∩I} f(t)dt - f(x)| < r$. Kempisty's theorem concerning the product quasicontinuity is investigated for the above notions.
In [4] W. Li and S.S. Cheng prove a Picard type existence and uniqueness theorem for iterative differential equations of the form y'(x) = f(x,y(h(x)+g(y(x)))). In this article I show some analogue of this result for a larger class of functions f (also discontinuous), in which a unique differentiable solution of considered Cauchy's problem is obtained.
4
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
A sequence (f n)n of functions f n: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f n+1(x) ≤ f n (x) (f n+1(x) ≥ f n(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.
5
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Let I ⊂ ℝ be an open interval and let A ⊂ I be any set. Every Baire 1 function f: I → ℝ coincides on A with a function g: I → ℝ which is simultaneously approximately continuous and quasicontinuous if and only if the set A is nowhere dense and of Lebesgue measure zero.
6
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We investigate functions f: I → ℝ (where I is an open interval) such that for all u,v ∈ I with u < v and f(u) ≠ f(v) and each c ∈ (min(f(u),f(v)),max(f(u),f(v))) there is a point w ∈ (u,v) such that f(w) = c and f is approximately continuous at w.
7
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW