Decomposing symmetrically continuous and Sierpinski-Zygmund functions into continuous functions

In this paper we will investigate the smallest cardinal number such that for any symmetrically continuous function there is a partition of such that every restriction is continuous. The similar numbers for the classes of Sierpinski-Zygmund functions and all functions from to are also investigated and it is proved that all these numbers are equal. We also show that and that it is consistent with ZFC that each of these inequalities is strict.

     

Axial continuous functions

A function from the plane to the plane is axial if it does not change one coordinate. We show that not every continuous function can be approximated by a superposition of continuous axial functions. This is a counterexample to a possible generalization of theorem of Eggleston about continuous bijections.     

Measurable functions and almost continuous functions

I show that if (X, ) is a Radon measure space and Y is a metric space, then a function from X to Y is -measurable iff it is almost continuous (=Lusin measurable). I discuss other cases in which measurable functions are almost continuous.Part of the work of this paper was done during a visit to Japan supported by the United Kingdom Science Research Council and Hokkaido University     

Hausdorff continuous interval-valued functions and quasicontinuous functions

In the paper it is shown that every Hausdorff continuous interval-valued function corresponds uniquely to an equivalence class of quasicontinuous functions. This one-to-one correspondence is used to construct the Dedekind order completion of C(X), the set of all real-valued continuous functions, when X is a compact Hausdorff topological space or a complete metric space.     

Families of meromorphic functions avoiding continuous functions

Leth ₁,h ₂ andh ₃ be continuous functions from the unit disk D into the Riemann sphereC such thath _i(z) ≠ h_j(z) (i ≠ j) for eachz∈D. We prove that the setF of all functionsf meromorphic on D such thatf(z)≠h _j (z) for allz ∈ D andj=1,2,3 is a normal family. The result and the method of the proof extend to quasimeromorphic functions in higher dimensions as well. The second author was supported by a Heisenberg fellowship of the DFG. The fourth author was partially supported by the Marsden Fund, New Zealand. This research was completed while the authors were attending a conference at Mathematisches Forschungsinstitut Oberwolfach in Germany. The authors would like to express their sincere thanks to the Institute for providing a stimulating atmosphere and for its kind hospitality.     

Sequences of semicontinuous functions accompanying continuous functions

A space X is said to have property (USC) (resp. (LSC)) if whenever is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [0,1] converging pointwise to the constant function 0 with the value 0, there is a sequence of continuous functions from X into [0,1] such that fn?gn (n∈ω) and converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S₁(Γ,Γ) and wQN.     

Restrictions of continuous functions

Let α = {α₁, ...,α_k} be a finite multiset of non-negative real numbers. Consider the sequence of all positive integer multiples of all α _i ’s, and note the multiplicity of each term in this sequence. This sequence of multiplicities is the resonance sequence generated by {α ₁, ...,α_k}. Two multisets are combinatiorially equivalent if they generate the same resonance sequence. The paper is devoted to the classification of multisets up to combinatorial equivalence. We show that the problem of combinatorial equivalence of multisets is closely related to the problem of classification of systems of second order ordinary differential equations up to focal equivalence.     

Glueing continuous functions constructively

The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect.     

Column continuous transition functions

A column continuous transition function is by definition a standard transition function P(t) whose every column is continuous for t?0 in the norm topology of bounded sequence space l_∞. We will prove that it has a stable q-matrix and that there exists a one-to-one relationship between column continuous transition functions and increasing integrated semigroups on l_∞. Using the theory of integrated semigroups, we give some necessary and sufficient conditions under which the minimal q-function is column continuous, in terms of its generator (of the Markov semigroup) as well as its q-matrix. Furthermore, we will construct all column continuous Q-functions for a conservative, single-exit and column bounded q-matrix Q. As applications, we find that many interesting continuous-time Markov chains (CTMCs), say Feller-Reuter-Riley processes, monotone processes, birth-death processes and branching processes, etc., have column continuity.     

Non continuous liapunov functions

Summary Theorems which give sufficient conditions for various kinds of qualitative behaviours of flows defined by ordinary differential equations are proved. These theorems are based upon suitable properties of non continuous real-valued functions and their lower-right-hand-side Dini derivatives along the trajectories of the system. Entrata in Redazione il 10 febbraio 1969. Lavoro eseguito nell'ambito del Gruppo N. 11 del C. N. R.     

Approximately continuous functions which are continuous almost everywhere

Acta Mathematica Hungarica -     

Best approximations of continuous functions by spline functions

An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function = ₁ (x),..., _r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector ₀, exact estimates are obtained of the norms f(x)-s (f, ₀; x)_C[0,1] and f (x)-s (f, ₀; x)_{L[0, 1]} on the function classes H Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.     

Uniform approximation of continuous functions by rational functions

Summary Viene data la condizione necessaria e sufficiente perchè le funzioni razionali di una variabile, aventi poli di ordine prefissato in assegnati punti del piano complesso, costituiscano un sistema completo iu C^o (0, 1). A Bruno Finzi nel suo70 ^mo compleanno. This research has been sponsored in part by the Aerospace Research Laboratoires through the European Office of Aerospace Research, OAR, United States Air Force, under Grant EOOAR-69-0066. Entrata in Redazione il 22 aprile 1970.     

Weighted algebras of continuous functions

We give conditions under which the weighted spaces CV(X) and CV₀ (X) are locally convex algebras of a certain type and proceed to a systematic study of these algebras. We show, in particular, that the spectrum of CV₀ (X) is always homeomorphic to X and we detemine the spectrum of CV(X).     

Optimization of lipschitz continuous functions

Mathematical Programming - This paper contains basic results that are useful for building algorithms for the optimization of Lipschitz continuous functionsf on compact subsets of E n . In this...     

Graphical convergence of continuous functions

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈f _λ 〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.