A general conception of a completion is developed with illustrations, primarily crafted from Euclidean material, accompanied by a comprehensive survey of characterizations of completeness encountered in analysis.

For any map $f$ from $\mathbb{R}$ to $\mathbb{R}$, if an interval $J$ contains all periodic points of period 1 and 2, then $f(f(J))$ contains all periodic points (and therefore contains the centre of $f$).

Let $G$ and $H$ be Polish groups and let $\pi\colon G\rightarrow H$ be a function. The automatic continuity problem is the following: assuming $\pi$ is a group homomorphism, find conditions on $G$, $H$, or $\pi$ which imply that $\pi$ is continuous. In this note, we initiate a study of a reverse problem: supposing $\pi$ is continuous, find conditions on $G$, $H$, or $\pi$ which imply that $\pi$ is a homomorphism. Herein, we treat the case $G=H=\mathbb{R}$.

We extend the discretization method of de Guzmán to the setting of general metric measure spaces with mild assumptions on their structures. This method allows one to relate the best constants in the weak type $(1,1)$ inequalities for the relevant centered and uncentered Hardy-Littlewood maximal operators with the analogous constants received by applying the maximal operators to sums of Dirac deltas rather than to $L^1$ functions.

We prove that a function $f(x,y)$ of real variables defined on a rectangle, having partial derivatives $f''_{xx}, f''_{yy}\in L_2([0,1]^2)$, has almost everywhere mixed derivatives $f''_{xy}$ and $f''_{yx}$.

Let $\lambda$ denote the probability Lebesgue measure on $\mathbb{T}^2$. For any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $\lambda$ with measure-theoretic entropy equal to topological entropy, we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). This generalizes the result of Tseng (2009) for $C^2$-expanding maps of the circle.

During the last few decades E. S. Thomas, S. J. Agronsky, J. G. Ceder, and T. L. Pearson gave an equivalent definition of the real Baire class 1 functions by characterizing their graphs. In this paper, using their results, we consider the following problem: let $T$ be a given subset of $[0,1]\times\mathbb{R}$. When can we find a function $f:[0,1]\rightarrow\mathbb{R}$ such that the accumulation points of its graph are exactly the points of $T$? We show that if such a function exists, we can choose it to be a Baire-2 function. We characterize the accumulation sets of bounded and not necessarily bounded functions separately. We also examine the similar question in the case of Baire-1 functions.

This paper descibes our hunt for the solver of Problem 157 in the Scottish Book, a problem originally posed by A. J. (Gus) Ward in 1937. We first make the observation that a theorem of Richard O’Malley from 1975 yields an immediate positive solution. A further look at O’Malley’s references revealed a 1970 paper by Donald Ornstein that we now believe contains the first solution of SB 157. We isolate the common elements in the machinery used by both Ornstein and O’Malley and discuss several consequences. We also examine an example function given by Ornstein. There are some difficulties with this function but we provide a fix, and show moreover that functions of that kind are typical in the sense of the Baire category theorem.

For each $s \in (0,1]$, we give an example of a nowhere dense perfect set $E$ contained in the unit interval with $\dim_{\cal H} (E) = s$, which is not an attractor for any iterated function system composed of weak contractions. This answers a problem posed by Zoltán Buczolich at the Summer Symposium in Real Analysis XXXIX (June 8-13, 2015, St.~Olaf College, Northfield, MN).

One solution to a relatively recent American Mathematical Monthly problem, requesting the evaluation of a real definite integral, could be couched in terms of a contour integral which vanishes a priori. While the required real integral emerged on setting to zero the real part of the contour quadrature, the obligatory, simultaneous vanishing of the imaginary part alluded to still another pair of real integrals forming the first two entries in the infinite log-sine sequence, known in its entirety. It turns out that identical reasoning, utilizing the same contour but a slightly different analytic function thereon, sufficed not only to evaluate that sequence anew, on the basis of a vanishing real part, but also, in setting to zero its conjugate imaginary part, to recover the fundamental Bernoulli number recurrence. The even order Bernoulli numbers $B_{2k}$ entering therein were revealed on the basis of their celebrated connection to Riemann,’s zeta function $\zeta(2k)$. Conversely, by permitting the related Bernoulli polynomials to participate as integrand factors, Euler’s connection itself received an independent demonstration, accompanied anew by an elegant log-sine evaluation, alternative to that already given. And, while the Bernoulli recurrence is intended to enjoy here the pride of place, this note ends on a gloss wherein all the motivating real integrals are recovered yet again, and in quite elementary terms, from the Fourier series into which the Taylor development for $\text{Log}(1-z)$ blends when its argument $z$ is restricted to the unit circle.

We prove that in any Sobolev space which is subcritical with respect to the Sobolev Embedding Theorem there exists a closed infinite dimensional linear subspace whose non zero elements are nowhere bounded functions. We also prove the existence of a closed infinite dimensional linear subspace whose non zero elements are nowhere $L^q$ functions for suitable values of $q$ larger than the Sobolev exponent.

This article presents an elementary proof of the Implicit Function Theorem for differentiable maps $F(x,y)$, defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base point. In the case of a single scalar equation this continuity hypothesis is not required. A stronger than usual version of the Inverse Function Theorem is also shown. The proofs rely on the mean-value and the intermediate-value theorems and Darboux’s property (the intermediate-value property for derivatives). These proofs avoid compactness arguments, fixed-point theorems, and integration theory.

We explore the properties of an interesting new example of a function which is Lebesgue integrable but not Riemann integrable.

In 1971, D. Ornstein proved a theorem that completely solved Problem 157 of the Scottish Book. The purpose of this paper is to give an independent proof.

In 1975, Richard O’Malley proved that every approximately continuous function has approximate extrema, and this result provides an immediate solution to SB 157. The purpose of this paper is to provide an additional proof of O’Malley's result.