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In this thesis, a continuous function is called a proper mean if it is symmetric, reflexive and monotonic. We mainly observe quasi-arithmetic and Lagrangian means. Lagrangian means are de ned as means obtained from applying the classical mean value formula to a continuous and strictly monotonic function. Quasi-arithmetic means are transformations of the common arithmetic mean. We de ne the quasiarithmetic mean Qf associated with f : I ! I as Qf (x; y) := f-1 f (x) + f (y) 2 x; y 2 I and Lagrangian mean Lf as Lf (x; y) := 8< : f1 1 yx Ry x f(t)dt ; if x 6= y; x; if x = y: Many well-known means like arithmetic and geometric mean are both Lagrangian and quasi-arithmetic. The harmonic mean however is not Lagrangian but it is quasi-arithmetic. It is shown that there is a close relationship between Lagrangian and quasi-arithmetic means. This relationship can be made explicit through a homeomorphism. The purpose of this thesis is to give answers for two questions: 1. Which means can be represented according to some function f as quasiarithmetic and Lagrangian mean? 2. How to describe function classes with the same quasi-arithmetic and Lagrangian mean? The answers are presented in the second and third chapter. 41 In the rst chapter, some of the well-known means are discussed. The de nition of a proper mean is given and illustrated with some examples. The second chapter of the thesis, is dedicated to quasi-arithmetic means. It is shown that Qf = Qg if and only if g = f + where ; 2 R and 6= 0: A proof of the famous Acz el theorem is given. In the last chapter of the thesis, Lagrangian means are observed. It is shown that Lf = Lg if and only if g = f + where ; 2 R and 6= 0.

The purpose of this thesis is to study the asymptotic behaviour of intermediate points in mean value theorems for integrals. The most simple mean value theorem states that if f : [a; b] ! R is a continuous function then there exists a number c 2 (a; b) such that Z b a f(t) dt = f(c)(b - a): In the case of the simpler mean value theorems for integrals the intermediate point c(x) asymptotically approaches the midpoint of the interval [a; x] and in addition lim sup x!a c(x) - a x - a 1 e : The simplest weighted mean value theorem for integrals states that if f : [a; b] ! R is a continuous function and g : [a; b] ! [0;1) is an integrable function then there exsists a number c 2 [a; b] such that Z b a f(t)g(t) dt = f(c) Z b a g(t) dt: In the case of the weighted mean value theorems for integrals the intermediate point asymptotically approaches the value a + (x - a) k q 1 k+1, where k 2 N is the number of times the function f di erentiable at the point a. Also when f and g are di erentiable then the intermediate point satis es the equation lim x!a R c(x) a g(t) dt R x a g(t) dt = 1 2 : 38 If the conditions set on the functions in the weighted mean value theorems for integrals are expanded to functions that aren't di erentiable at the point a then the approximate value c(x) a + (x - a) r r s + 1 r + s + 1 ; where r 2 (-1; 0) [ (0;1), s 2 (-1;1) and r +s > -1, can be used to provide close approximiations to certain physics' problems.

Uniform continuity of real functions on unbounded intervals is not a compulsory topic in the basic analysis courses. In the course «Calculus III», the study of the uniform continuity is limited to Cantor’s theorem. The theorem states that if f : [a; b] ! R is a continuous function, then it is uniformly continuous. This result is essential for proving one of the central results of the integral calculus, which states that if a function f : [a; b] ! R is continuous, then it is integrable on [a; b]. We also know that if f : [a;1) ! R is continuous and limx!1 f(x) exists (as a real number), then f is uniformly continuous on the interval [a;1). The aim of this bachelor thesis is to characterize functions f : [a;1) ! R that are uniformly continuous on [a;1). This thesis consists of three chapters and is based on the articles by R. L. Pouso [4] and S. Djebali [1]. In the first chapter of this bachelor thesis, we discuss the definition of a uniform continuity and bring some examples of continuous functions that are not uniformly continuous. In this introductory chapter we use sequences to investigate uniform continuity.We provide an important proposition on the relation between uniformly continuous function and Cauchy sequences, which we will use for further proofs. The second chapter is devoted to examining uniformly continuous functions on unbounded intervals. We present conditions for a function f : [a;1) ! R to be uniformly continuious on an interval [a;1). Then we give an example that shows that the condition limx!1 jf(x)j x = 0 does not guarantee uniform continuity of the function f. At the end of the chapter, we show that if the graph of a function f has a horizontal or oblique asymptote then the function is uniformly continuous. In the third chapter, the relation between uniform continuity and convergence of improper integrals is studied.We examine unifom continuity of a function f : [a;1) ! R, for which improper integral R 1 a f(x)dx convergences. We also provide example of cases when uniform continuity of a function does not guarantee integral convergence.

Ordinary differentiable functions are known from base course of calculus. In this bachelor thesis they are presented in a generalized meaning. The function f : D !R, where D " R, is said to be symmetrically differentiable at a point x #D if the limit lim h!0 f (x +h)− f (x −h) 2h := f s (x) exists. f s (x) is called the symmetric derivative of the function f . It is assumed that with some ! > 0 (x −!,x)%D &=! and (x,x +!)%D &=!. Denote that symmetrical differention is weaker condition than ordinary differention so, if f (a) exists then f s (a) = f (a). It is also seen that symmetrically differentiable function may not be continuous. In this paper mean value theorems of symmetrically differentiable functions are examined. They are derived from Lagrange’s, Rolle’s and Cauchy’s mean value theorems. This work consists of four chapters. In the first chapter there are necessary notions and results. Symmetrical continuity and differentiability are defined, connections with ordinary continuous and differentiation are found. It is proven that equations of symmetrically differentiable functions are similar to the equations of orinary differentiable functions. In the second chapter the main result — the mean value theorem for symmetrically differentiable functions — is presented. In addition the necessary conditions for this quasi–mean value theorem to manifest as classical one are found. In the second part of the chapter some applications of the quasi–mean value theorem are shown, the conditionswhen symmetrically differentiable functions are differentiable in ordinary sense, investigated. The third chapter introduces different versions of the quasi–mean value theorem. They are analogous to theorems for symmetrically differentiable functions of Flett, Trahan and others . In the last, fourth, chapter there is an overview of Cauchy like mean value theorems for symmetrically differentiable functions. They are based onWachnicks result, which is one of the versions of mean value theorem of Cauchy. This bachelor thesis is referable, it is drawn on the next articles: C. E. Aull [1], R. M.Davitt, R. C. Powers, T. Riedel, P. K. Sahoo [2], T. M. Flett [3], S. Reich [4], P. K. Sahoo [5], P. K. Sahoo [6], D. H. Trahan [8], E.Wachnick [9].