COMPOSITIONS OF DARBOUX-LIKE FUNCTIONS

Let \\(\\mathcal{F}\\) be a family of real functions, \\(\\mathcal{F}\\subseteq\\mathbb{R}^\\mathbb{R}\\). In the paper we will examine the following question. For which families \\(\\mathcal{F}\\subseteq\\mathbb{R}^\\mathbb{R}\\) does there exist \\(g\\colon\\mathbb{R}\\to\\mathbb{R}\\) such that \\(f+g\\in\\mathcal{F}\\) for all \\(f\\in \\mathcal{F}\\)? More precisely, we will study a cardinal function \\(\\text{A}(\\mathcal{F})\\) defined as the smallest cardinality of a family \\(F\\subseteq\\mathbb{R}^\\mathbb{R}\\) for which there is no such \\(g\\). We will prove that \\(\\text{A}(\\text{Ext})=\\text{A}(\\text{PR})=\\mathcal{c}^+\\) and \\(\\text{A}(\\text{PC})=2^{\\mathcal{c}}\\), where \\(\\text{Ext}\\), \\(\\text{PR}\\) and \\(\\text{PC}\\) stand for the classes of extendable functions, functions with perfect road and peripherally continuous functions from \\(\\mathbb{R}\\) into \\(\\mathbb{R}\\), respectively. In particular, the equation \\(\\text{A}(\\text{Ext})=\\mathcal{c}^+\\) immediately implies that every real function is a sum of two extendable functions. This solves a problem of Gibson \\cite{Gib1}. We will also study the multiplicative analogue \\(\\text{M}(\\mathcal{F})\\) of the function \\(\\text{A}(\\mathcal{F})\\) and we prove that \\(\\text{M}(\\text{Ext})=\\text{M}(\\text{PR})=2\\) and \\(\\text{A}(\\text{PC})=\\mathcal{c}\\). This article is a continuation of papers \\cite{N2,CM,NR} in which functions \\(\\text{A}(\\mathcal{F})\\) and \\(\\text{M}(\\mathcal{F})\\) has been studied for the classes of almost continuous, connectivity and Darboux functions.

First Return Approachability

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

In this paper we consider a cardinal invariant related to adding real functions defined on the real line. Let \(\mathcal{F}\) be such a family, we consider the smallest cardinality of a family \(\mathcal{G}\) of functions such that \(h+\mathcal{G}\) has non-empty intersection with \(\mathcal{F}\) for every function \(h\). We note that this cardinal is the additivity, a cardinal previously studied, of the compliment of \(\mathcal{F}\). Thus, we calculate the additivities of the compliments of various families of functions including the Darboux, almost continuous, extendable and perfect road functions. We briefly consider the general relationship between the additivity of a family and its compliment.

Darboux functions in the first class of Baire

It is known that the first return continuous functions are precisely the Darboux functions in Baire class 1, and that every such function can be changed via a homeomorphism into an approximately continuous function. Here we give two characterizations of the smaller class of universally first return continuous functions, one of which is the capacity of changing such a function via a homeomorphism into an approximately continuous function which is continuous almost everywhere.

It is known that the first return continuous functions are precisely the Darboux functions in Baire class 1, and that every such function can be changed via a homeomorphism into an approximately continuous function. Here we give two characterizations of the smaller class of universally first return continuous functions, one of which is the capacity of changing such a function via a homeomorphism into an approximately continuous function which is continuous almost everywhere.

The theory of functions of one or more real variables grew out of the attempt to understand and clarify a number of strange discoveries that had been made in the nineteenth century. Continuous but non-differentiable functions, series of continuous functions whose sum is discontinuous, continuous functions that are not piecewise monotonic, functions possessing bounded derivatives that are not Riemann integrable, curves that are rectifiable but not according to the calculus definition of arc length, and nonintegrable functions that are limits of sequences of integrable functions-all seemed to contradict the expected behavior of functions, derivatives, and integrals. Another motivation for the further study of the behavior of functions came from the research on Fourier series. This theory, as built up by Dirichlet, Riemann, Cantor, Ulisse Dini (1845-1918), Jordan, and other mathematicians of the nineteenth century, was a quite satisfactory instrument for applied mathematics. But the properties of the series, as thus far developed, failed to give a theory that could satisfy the pure mathematicians. Unity, symmetry, and completeness of relation between function and series were still wanting.

On the universal completion of pointfree function spaces

Let D be the open unit disk in the complex plane, and let C be its boundary, the unit circle.If x £ C, then by an arc at x we mean a simple arc y with one end point at x such that y-{x}£ö.If/is a function mapping D into some metric space M, then the set of curvilinear convergence of/is defined to be {x e C: there exists an arc y at x and there exists a point p e M such that f(z) ->p as z -»■ x along y}.If ^ is a function whose domain is a subset F of the set of curvilinear convergence of/ then <f> is called a boundary function for/if, and only if, for each x e E there exists an arc y at x such that/(z) -*■ <f>(x) as z -> x along y.Let 5 be another metric space.We shall say that a function <f> is of Baire class ^ 1(5, M) if (i) domain <£ = S, (ii) range ^ £ M, and(iii) there exists a sequence {<£"} of continuous functions, each mapping S into M, such that <f>n^>-<p pointwise on 5. We shall say that (f> is of honorary Baire class ^ 2(S, M) if (i) domain <f> = S, (ii) range ^£M, and (iii) there exists a countable set TVs 5 and there exists a function y!r of Baire class ^ 1(5, M) such that <f>(x) = tp(x) for every xe S-N.It is known that if/is a continuous function mapping F into the Riemann sphere, then the set of curvilinear convergence off is of type Fa6, and any boundary function for/is of honorary Baire class ^2(C, Riemann sphere).(See [3], [4], [5], [6],[9].) J. E. McMillan [6] posed the following problem.If A is a given set in C of type Fao, and if </> is a function of honorary Baire class Ú2(A, Riemann sphere), does there always exist a continuous function / mapping D into the Riemann sphere such that A is the set of curvilinear convergence of/ and <f> is a boundary function for /?The purpose of this paper is to give an affirmative answer to McMillan's question.However, the corresponding question for real-valued functions remains open.(See Problems 1 and 2 at the end of this paper.)In proving our result, we first give a proof under the assumption that <j> is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere.As we shall indicate in an appendix, the same device can be

We establish dichotomy results concerning the structure of Baire class 1 functions. We consider decompositions of Baire class 1 functions into continuous functions and into continuous functions with closed domains. Dichotomy results for both of them are proved: a Baire class 1 function decomposes into countably many countinuous functions, or else contains a function which turns out to be as complicated with respect to the decomposition as any other Baire class 1 function; similarly for decompositions into continuous functions with closed domains. These results strengthen a theorem of Jayne and Rogers and answer some questions of Steprāns. Their proofs use effective descriptive set theory as well as infinite Borel games on the integers. An important role in the proofs is played by what we call, in analogy with being Wadge complete, complete semicontinuous functions. As another application of our study of complete semicontinuous functions, we generalize some recent theorems of Jackson and Mauldin, and van Mill and Pol concerning measures viewed as examples of complicated semicontinuous functions. We also prove that a Borel set A A is either Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } or there is a continuous injection ϕ : ω ω → A \phi :\; \omega ^{\omega }\to A such that for any Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } set B ⊂ A B\subset A , ϕ − 1 ( B ) \phi ^{-1}(B) is meager. We show analogous results for Borel functions. These theorems give a new proof of a result of Stern, strengthen some results of Laczkovich, and improve the estimates for cardinal coefficients studied by Cichoń, Morayne, Pawlikowski, and the author.

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

On the Kolmogorov complexity of continuous real functions

This paper introduces the frame of partially defined real numbers and the lattice-ordered ring of partial real functions on a frame. This is then used to construct the order completion of rings of pointfree continuous real functions. The bounded and integer-valued cases are also analysed. The application of this pointfree approach to the classical case C(X) of the ring of continuous real-valued functions on a topological space X yields a new construction for the Dedekind completion of C(X), considerably more direct and natural than the known procedure using Hausdorff continuous functions.

JOHN R. RICE(')1.1.Let two real functions/ and F be given such that/=/(x) is a continuous function for x£ [0, l] and F=F(al, a2, ■ ■ ■ , a", x) is a continuous function of n parameters and x(E [0, l].For simplicity of notation the point (a1, a2, ■ ■ ■ , a") in Euclidean w-space, En, is denoted by a.The domain of the parameters of F is denoted by P, a subset of En.The main problem in the theory of approximation of continuous functions may be stated as follows: Determine a*EP so that the deviation of the function F(a, x) from f(x) shall be minimized.Naturally one must define the deviation of F(a, x) from/(x) and different definitions lead to different theories.In this paper the deviation of F(a, x) from f(x) is taken to be maxI(=[o,i] | F(a, x) -f(x) \.All maxima and minima are taken over x£ [0, l] unless otherwise stated.F(a*, x) is said to be a best approximation to/The results of this paper remain valid if [0, l] is replaced by any other closed and bounded interval.By 1920 an elegant theory of Tchebycheff approximation had been developed for F(a, x) depending linearly on the parameters [2; 3; 7].Some of the main points of this linear theory are given in the following resume.Let T be a set of n functions 4>i(x), 4>2(x), • • • , </>"(x) continuous on [0, 1 ] and let P(a, x) = /,?_, a*'<£,(x) be called a T-polynomial.P(a, x) is said to be a nontrivial T-polynomal if 2~ll=i | a*| >0.T is said to be a Tchebycheff Set if every nontrivial polynomial has at most n-1 zeros, max | F(a, x) -/(x)| is said to alternate n times if there are « + l points 0^Xi<x2<Alternance is a basic concept in the theory of Tchebycheff approximations.The following theorem answers the principal questions of a general nature in the theory of linear Tchebycheff approximations.Theorem.Let T be a Tchebycheff set and let f(x) be an arbitrary function continuous on [0, l].Then A. f(x) possesses a best approximation, B. a necessary and sufficient condition that P(ao, x) be a best approximation to f(x) is that max | P(a0, x) -f(x) \ alternates at least n times, C. the best approximation to f(x) is unique.