Inequalities connected with the moduli of smootness
Insertion of a Contra-Baire- (Baire-) Function
Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- function between two comparable real-valued functions on the topological spaces that -kernel of sets are -sets.
Kempisty's theorem for the integral product quasicontinuity
A function f: ℝⁿ → ℝ satisfies the condition (resp. , ) 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 . Kempisty’s theorem concerning the product quasicontinuity is investigated for the above notions.
Kolmogorov problem in and extremal Zolotarev ω-splines [Book]
Le rang de Baire de la famille de toutes les fonctions ayant la propriété (K)
Les ensembles de niveau et la monotonie d'une fonction
Les fonctions qui ont la propriété (K) et la mesurabilité des fonctions de deux variables
L’espace linéaire des fonctions cliquées sur est généré par les fonctions quasi-continues
Limits of approximately continuous functions
Linear operators in the space of regulated functions
Representation of bounded and compact linear operators in the Banach space of regulated functions is given in terms of Perron-Stieltjes integral.
Maximums of Darboux quasi-continuous functions
Maximums of strong Świątkowski functions
Micro tangent sets of continuous functions
Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s...
Monotonicity, continuity and levels of Darboux functions
Monotonicity of certain functionals under rearrangement
We show here that a wide class of integral inequalities concerning functions on can be obtained by purely combinatorial methods. More precisely, we obtain modulus of continuity or other high order norm estimates for functions satisfying conditions of the type where and are monotone increasing functions of .Several applications are also derived. In particular these methods are shown to yield a new condition for path continuity of general stochastic processes
More on the Continuity of Real Functions
In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.
“More or less" first-return recoverable functions.
Multiplying balls in the space of continuous functions on [0,1]
Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set is residual whenever E is residual in...
New types of continuities.
Nonlinear contractive conditions: A comparison and related problems
We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations,...