Properties of Peripherally Continuous Functions and Connectivity Maps

Mathematic

Cardinal Invariants Concerning Extendable and Peripherally Continuous Functions

Let F be a family of real functions, F ⊆ R R . In the paper we will examine the following question. For which families F ⊆ R R does there exist g : R → R such that f + g ∈ F for all f ∈ F? More precisely, we will study a cardinal function A(F) defined as the smallest cardinality of a family F ⊆ R R for which there is no such g. We will prove that A(Ext) = A(PR) = c + and A(PC) = 2c , where Ext, PR and PC stand for the classes of extendable functions, functions with perfect road and peripherally continuous functions from R into R, respectively. In particular, the equation A(Ext) = c + immediately implies that every real function is a sum of two extendable functions. This solves a problem of Gibson [6]. We will also study the multiplicative analogue M(F) of the function A(F) and we prove that M(Ext) = M(PR) = 2 and A(PC) = c. This article is a continuation of papers [10, 3, 12] in which functions A(F) and M(F) has been studied for the classes of almost continuous, connectivity and Darboux functions

Darboux Like Functions that are Characterizable by Images, Preimages and Associated Sets

For arbitrary families A and B of subsets of R let C(A,B)= {f| f: R--\u3eR and the image f[A] is in B for every A in A} and C-1 (A,B)= {f| f: R--\u3eR and the inverse image f-1(B) is in A for every B in B}. A family F of real functions is characterizable by images (preimages) of sets if F=C(A,B) (F=C-1(A,B), respectively) for some families A and B. We study which of classes of Darboux like functions can be characterized in this way. Moreover, we prove that the class of all Sierpinski-Zygmund functions can be characterized by neither images nor preimages of sets

The largest linear subspace contained in Darboux-likr functions on R

Consider an arbitrary $\mathcal F\subset\mathbb R^\mathbb R$, where the family $\mathbb R^\mathbb R$ of all functions from $\mathbb R$ to $\mathbb R$ is considered as a linear space over $\mathbb R$. Does $\mathcal F\cup\{0\}$ contain a non-trivial lineal subspace? If so, how big the vector space can be? These questions are at the core of lineability theory. In particular, we say that a family $\mathcal F\subset\mathbb R^\mathbb R$ is {\em lineable (in $\mathbb R^\mathbb R$)}\/ provided there exists an infinite dimensional linear space contained in $\mathcal F\cup\{0\}$. There has been a lot of attention devoted to lineability problem of subsets of linear space of functions. For instance, the families of continuous nowhere differentiable functions and of differentiable nowhere monotone functions are lineable. It has also been known for a while that the class $D\subset\mathbb R^\mathbb R$ of Darboux functions (i.e., functions that satisfy the intermediate value property) is lineable. In fact, $D$ is $2^\mathfrak c$-lineable, that is, $D\cup\{0\}$ contains a subspace of dimension $2^\mathfrak c$, where $2^\mathfrak c$ is the cardinality of $\mathbb R^\mathbb R$. The goal of this work is to study the lineabilitiy of the subclasses of $D$ that are in the algebra generated by $D$ and seven of its subclasses (known as Darboux-like functions): extendable ($Ext$), almost continuous ($AC$), connectivity ($Conn$), peripherally continuous ($PC$), having perfect road ($PR$), having Cantor Intermediate Value Property ($CIVP$), and having Strong Cantor Intermediate Value Property ($SCIVP$). This dissertation is arranged as follows. Chapter~1 focuses on presenting notations, definitions, and summary of all results contained in this work. In chapter~2, we give a general method to have $\mathfrak c$-lineable for all Darboux-like maps and even their restriction to Baire 2 class functions. In chapter~3, we will build some tools that allow us to show $2^\mathfrak c$-lineability (i.e., maximal lineability) for all Darboux-like subclasses of $(PC\setminus D)\cup(AC\setminus Ext)$ in the algebra. In chapter~4, we are going to construct algebraically independent sets that shall be used to achieve the maximal lineability for all Darboux-like subclasses of $D\setminus Conn.$ In the last part, in chapter~5, we will make some remarks on lineability and offer new possibilities for open problems

Set Theoretic Real Analysis

This article is a survey of the recent results that concern real functions (from Rn into R) and whose solutions or statements involve the use of set theory. The choice of the topics follows the author\u27s personal interest in the subject, and there are probably some important results in this area that did not make to this survey. Most of the results presented here are left without the proofs

On Peano spaces, with special reference of unicoherence and non-continuous functions

We have mentioned that each chapter in this thesis is conceived of as an independent paper, except for Chapter 3, which is a collection of results on non- continuous functions. Consequently each chapter contains a clearly marked introductory section, in which its back- ground and content are explained. In this abstract we shall summarize the remarks in these introductory sections. In chapter 1 we present an n-arc theorem for Peano spaces which is an extension of the theorem in §2 of [32], which Menger called the second n-arc theorem in [17]. Whereas in the second n-arc theorem n disjoint arcs are constructed joining two disjoint closed sets A and B, in chapter 1 we split the closed set A into n dis- joint closed subsets A1 A 2, ••• , An and give necessary and sufficient conditions for there to be n disjoint arcs joining A and B, one meeting each A1. At the end of chapter 1 we present a conjecture, which we have been able to verify in special cases. In [35] Whyburn proved a theorem concerning the weak connected separation of two non-degenerate connected closed sets A and B by a quasi-closed set L in a locally cohesive space X. In chapter 2 we show that A and B can in fact be taken as arbitrary closed sets in this theorem; that is, ,we remove the restriction of non-degeneracy and connectedness on A and B. In chapter 3 we study the circumstances under which a connectivity function is peripherally continuous. The study of the abstract relations between non- continuous functions was initiated by Stallings in [23]. In this paper he introduced the 1pc polyhedron and showed that a connectivity function was peripherally continuous on an 1pc polyhedron. Whyburn took up the study of non- continuous functions in [33]. [34] and [35]. He introduced the locally cohesive space, which is more general than the 1pc polyhedron, and proved that a connectivity function was peripherally continuous on a locally cohesive Peano space. For technical reasons, the locally cohesive space is not permitted to have local cut points. It is obvious, however, that on many Peano spaces having local, cut points a connectivity function remains peripherally continuous, In §2,3 of chapter 3 we formulate a sequence of properties Pn(X), which permit the space X to have local cut points, and we prove in each case that a connectivity function f : X →Y is peripherally continuous when X has property Pn(X). Each of these properties is an improvement on the last, and the final one, the U-space, satisfactorily incorporates the class of Peano spaces with local cut points on which we are able to prove that a connectivity function is peripherally continuous. An interesting feature of §3 of chapter 3 is provided by two "weak separation theorems," and more will be found about these in the introduction to chapter 3. In §4 of chapter 3 we show that a connectivity function is peripherally continuous on a locally compact ANR. This affirmatively answers a question that Stallings raised in [23]. The U-space that we have introduced in §3 of chapter 3 imposes a "unicoherence condition" in the space X (as do all the properties Pn(X) considered in §3, chapter 3). In §5 of chapter 3 we generalize the U-space to the S-space. This imposes a "multicoherence condition" on the space X, and we prove that a connectivity function is peripherally continuous on a cyclic S-space. We close chapter 3 by considering the question of placing weaker conditions than connectivity on the function f : X → Y which will still ensure that f is peripherally continuous. It is well known that if X is a unicoherent Peano continuum and A1, A 2, … is a sequence of disjoint closed subsets of X no one of which separates X, then Un=1 An does not separate X. In [28] van Est proved this theorem for the case where X is a Euclidean space of n dimensions. In chapter 4 we give an example which shows that this theorem does not hold if X is an arbitrary Peano space • In chapter 5 we provide a new angle to Lebesgue's covering lemma. We show that if the Lebesgue number ᵹ of an open covering U1, U2, •••• Un of a compact metric space X. ρ is finite. then it can be defined by the formula ᵹ = min ρ (E, F), where E and F are any compartments contained in no common U1 In chapter 6 we show that an involution on a cyclic Peano space leaves some simple closed curve setwise invariant. Whyburn has given a proof of R. L. Moore’s decomposition theorem for the 2-sphere in [31] (a refinement of this proof is presented in [36]). His proof is accomplished by showing that the decomposition space satisfies Zippin’s characterization theorem for the 2-sphere. In chapter 6 we present an alternative way of showing that the decomposition space satisfies Zippin's characterization theorem. Our proof closely follows Alexander's proof of the Jordan curve theorem as given by Newman in[21], and so consists of arguments that are well-known in another context. In [30] Whyburn gave a proof of the cyclic connectivity theorem. and in all subsequent appearances of this theorem in the literature Whyburn's proof has been used. Whyburn divided the proof of the theorem into three parts lemma 1, lemma 2, and the deduction of the theorem from lemmas 1 and2. In chapter 8 we give an alternative proof of lemma 1. Our proof is based on the fact that a cyclic Peano space has a base of regions whose closures do not separate the space, and it proceeds by an induction on a simple chain of these regions

Functional Organization of the Gustatory System in the Brains of Ictalurid Catfish: a Combined Electrophysiological and Neuroanatomical Study (Taste, Viscerotopic, Sensory Maps, Forebrain).

The present study utilizes electrophysiological and neuroanatomical techniques to investigate the functional organization of the gustatory system in the brainstem and the forebrain of the channel catfish, Ictalurus punctatus. Neuroanatomical studies indicate an overlapping, segmental pattern of projection of glossopharyngeal-vagal branches in the vagal lobe. The vagal nerve complex is divisible into an interoceptive input (consisting of general visceral fibers) from abdominal viscera and an exteroceptive-branchial component (consisting of special and general visceral fibers) innervating the oro-pharyngeal region. The interoceptive-visceral input converges onto the exteroceptive, oro-pharyngeal input in the nucleus intermedius of the vagal lobe (nIV). In addition, extra-oral and oral gustatory information converges onto the nucleus intermedius of the facial lobe (nIF) and sensory inputs from separate regions of the oropharynx converge onto separate halves of the dorsal cap of the vagal lobe. Overlapping taste and tactile sensory maps of the oropharynx are present in the vagal lobe of the catfish. The representation of the oropharynx is less well defined than the somatotopic map in the facial lobe except for the bilaterally mapped extra-oral surface. Gustatory information reaches the area dorsalis pars medialis of the telencephalon and several nuclei in the ventral diencephalon of the catfish. The central gustatory pathway ascends from the medulla to the level of the diencephalon via the secondary gustatory nucleus as well as to the telencephalon via small neurons in the diencephalic lobo-bulbar nucleus. Neurons in the gustatory region of the telencephalon descend to the diencephalic level primarily via the medial forebrain bundle

Additivity coefficients for all classes in the algebra of Darboux-Like maps on R

The class D of generalized continuous functions on R known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property or the Strong Cantor Intermediate Value Property. The algebra A(D) of classes of functions generated by these families contains 17 atoms. In this work we will calculate the values of the additivity coefficient A(F) for all atoms F in the algebra A(D). We also determine the values A(F) for a lot of other families F∈A(D). Open questions and new directions of research shall also be provided