A history of the Baire Category Theorem and numerous examples of its use in real analysis.

We prove a version of Schur’s lemma for operators with positive kernels on weighted \(L_p\) spaces and apply the result to Riesz potentials of first order to get weighted generalizations of Trudinger’s limiting embedding.

Two characterizations of Darboux Baire 1 functions are given: one in terms of pointwise limits of certain continuous functions, and the other in terms of intersections of sequences of open sets of a particular type.

Let \(0\lt p\lt q\lt \infty\). We investigate the following question: For which subsets \(H\) of the circle group \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\) is it true that if \(f\in L_p\) and \(\Delta_h f(x)=f(x+h)-f(x)\in L_q\) for any \(h\in H\), then \(f\in L_q\)? We prove that this is not true for pseudo-Dirichlet sets. Evidence is gathered for the conjecture that the class of counter-examples is precisely the class of \(N\)-sets.

For \(\mathcal{A},\mathcal{B}\subset\mathcal{P}(\mathbb{R})\) let \(\mathcal{C}_{\mathcal{A},\mathcal{B}}=\{ f\in\mathbb{R}^\mathbb{R}\colon(\forall A\in\mathcal{A})\,(f(A)\in\mathcal{B})\}\) and \(\mathcal{C}_{\mathcal{A},\mathcal{B}}^{-1}=\{ f\in\mathbb{R}^\mathbb{R}\colon(\forall B\in\mathcal{B})\,(f^{-1}(B)\in\mathcal{A})\}\). A family \(\mathcal{F}\) of real functions is characterizable by images (preimages) of sets if \(\mathcal{F}=\mathcal{C}_{\mathcal{A},\mathcal{B}}\) (\(\mathcal{F}=\mathcal{C}_{\mathcal{A},\mathcal{B}}^{-1}\), respectively) for some \(\mathcal{A},\mathcal{B}\subset\mathcal{P}(\mathbb{R})\). We study which of the classes of Darboux like functions can be characterized in this way. Moreover, we prove that the class of all Sierpiński-Zygmund functions can be characterized by neither images nor preimages of sets.

In this paper we consider the following questions: Under what additional assumptions is the diagonal \(f\triangle g\) of almost continuous functions vanishing on the set of discontinuity points of an almost continuous function, too. Moreover, one can show that the facts considered (diagonals and ac-homotopies) can be applied to the characterization of continuity or the investigations of the algebraic operations on almost continuous functions.

Some properties of harmonic functions defined outside a compact set in \({\mathbb R}^n\) are given. From them is deduced a generalized form of Liouville’s theorem in \({\mathbb R}^n\) which is known to be equivalent to an improved version of the classical Bôcher theorem on harmonic point singularities.

In this paper we show that Lusin’s condition \((N)\) and Foran’s condition \((M)\) are equivalent for Borel functions that are \(VBG\) on a Borel set. Also new characterizations of conditions \((M)\) and \(\underline{M}\) are given.

In this paper we shall study three kinds of descriptive type integrals, that all generalize the wide Denjoy integral. In fact the classes of primitives for these integrals, restricted to the continuous functions, are \(ACG\). We shall also study five kinds of Perron type integrals, that are all in a close relationship with the descriptive type integrals. In the last three sections we show some relationships between the descriptive type integrals and the Perron type integrals.

We consider local dimensions of probability measure on a complete separable metric space \(X\): \(\overline{\alpha}_\mu(x) =\underset{r\to 0}{\varlimsup} \frac{\log_\mu(B_r(x))}{\log r}, \underline{\alpha}_\mu(x)=\underset{r\to 0}{\varliminf} \frac{\log_\mu(B_r(x))}{\log r}\). We show (Theorem 2.1) that for a typical probability measure \(\underline{\alpha}_\mu(x)=0\) and \(\overline{\alpha}_\mu(x)=\infty\) for all \(x\) except a set of first category. Also \(\underline{\alpha}_\mu(x)=0\) almost everywhere and with some additional conditions on \(X\) there is a corresponding result for upper local dimension: in particular, we show that a typical measure on \([0,1]^d\) has \(\overline{\alpha}_\mu(x)=d\) almost everywhere (Theorem 2.4). There are similar results concerning ``global'' dimensions of probability measures. Theorems 2.2 and 2.3 show in particular that the Hausdorff dimension of a typical measure on any compact separable space equals 0 and the packing dimension of a typical measure on \([0,1]^d\) equals \(d\).

Constructions of homeomorphisms on subsets of \(\mathcal{R}\) are constructed and a conjecture is stated about the existence of such homeomorphisms.

We estimate the Hausdorff dimension of some Borel sets determined by the digits in triadic expansions.

In this note we will show that many classes \(\mathcal{F}\) of real functions \(f\colon {\mathbb R}\to\mathbb{R}\) can be characterized by preimages of sets in a sense that there exist families \(\mathcal{A}\) and \(\mathcal{D}\) of subsets of \(\mathbb{R}\) such that \(\mathcal{F}=\mathcal{C}(\mathcal{D},\mathcal{A})\), where \(\mathcal{C}(\mathcal{D},\mathcal{A})=\{f\in\mathbb{R}^\mathbb{R}\colon f^{-1}(A)\in \mathcal{D}\ \text{ for every } A\in\mathcal{A}\}.\) In particular, we will show that there exists a Bernstein \(B\subset \mathbb{R}\) such that the family \(\Delta\) of all derivatives can be represented as \(\Delta=\mathcal{C}(\mathcal{D},\mathcal{A})\), where \(\mathcal{A}=\bigcup_{c\in\mathbb{R}}\{(-\infty,c),(c,\infty),B+c\}\) and \(\mathcal{D}=\{g^{-1}(A)\colon A\in\mathcal{A}\ \&\ g\in\Delta\}\).

An example is constructed under the Continuum Hypothesis showing that almost continuity and the Strong Cantor Intermediate Value Property do not imply extendability. This answers a question in \cite{R11}. Results about stationary sets are given for the class of extendable functions from \(I\) into \(I\), where \(I = [0,1]\).

We introduce the condition \((PAC^*)\) that is a slight modification of the condition \((PAC)\) of Sarkhel and Kar. The main result is Theorem 4: \emph{A function \(f:[a,b] \to {\mathbb R}\) is \(VB^*G ąp (N)\) on a subset \(E\) of \([a,b]\) if and only if \(f \in (PAC^*)\) on \(E\)}. Consequently, \emph{the set \(\{f:[a,b] \to {\mathbb R}: f \in VB^*G ąp (N)\) on \(E\}\) is an algebra, whenever \(E\) is a subset of \([a,b]\)}. Using Theorem 1, we find seven characterizations of \(VB^*G ąp (N)\) on a Lebesgue measurable set (Theorem 5). We also give fifteen characterizations of the class of \(AC^*G\) functions on a closed set \(E\), that are continuous at each point of \(E\) (Theorem 6). In the last two sections, using Thomson’s outer measure \({\mathcal S}_o\text{-}\mu_f\), we characterize a \(VB^*G ąp (N)\) function \(f\) on a Lebesgue measurable set (Theorem 9). As a consequence we obtain that: \emph{A function \(f:[a,b] \to {\mathbb R}\) is \(AC^*G\) on a closed subset \(E\) of \([a,b]\) and continuous at each point of \(E\) if and only if \({\mathcal S}_o\text{-}\mu_f(Z) = 0\) whenever \(Z\) is a null subset of \(E\)} (Theorem 10).

Some special conditions (equidifferentiability or absolute equicontinuity) implying (or not) the differentiability of functions of two variables are considered.

We show that \(VBG \cap (N)\) is equivalent with Sarkhel and Kar’s class \((PAC)G\) on an arbitrary real set. Hence \(VBG \cap (N)\) is an algebra on that set. In Theorem 4, we give three characterizations for \(VBG \cap (N)\) on an arbitrary real set. It follows that Gordon’s \(AK_N\)-integral is a special case of the \(PD\)-integral of Sarkhel and De (Remark 3). In Theorem 3 we obtain the following surprising result: a Lebesgue measurable function \(f\) is \(VBG\) on \(E\) if and only if \(f\) is \(VBG\) on any null subset of \(E\). We also find seven characterizations of \(VBG ąp (N)\) for Lebesgue measurable functions (see Theorem 5). For continuous functions on a closed set, we obtain several characterizations of the class \(ACG\). Using a different technique, we obtain other characterizations of \(VBG \cap (N)\) for a Lebesgue measurable function (see Theorem 8).

Let \(C(f)\) denote the set of points at which a function \(f:I\to I\) is continuous, where \(I=[0,1]\). We show that if a Darboux quasicontinuous function \(f\) has a graph whose closure is bilaterally dense in itself, then \(f\) is extendable to a connectivity function \(F: I^2\to I\) and the set \(I\setminus C(f)\) of points of discontinuity of \(f\) is \(f\)-negligible. We also show that although the family of Baire class 1 quasicontinuous functions can be characterized by preimages of sets, the family of Darboux quasicontinuous functions cannot. An example is found of an extendable function \(f: I\to \mathbb{R}\) which is not of Cesaro type and not quasicontinuous.

Let \(g\) be a measurable real valued function on a bounded, measurable subset of the real line. We prove that if \(g(E)\) has measure 0, then 0 is one of the derived numbers of \(g\) at almost every point in \(E\). We find a function \(H\) on the real line that is nondecreasing and closely associated with \(G\), such that if \(g(E)\) has measure 0, the \(H'\) vanishes almost everywhere. Moreover, if \(g\) is an \(N\)-function on \(E\) and if \(H'\) vanishes almost everywhere, then \(g(E)\) has measure 0.

If a subset \(A\) of the real line is a countable union of closed, strongly symmetrically porous sets, then there exists a Lipschitz everywhere symmetrically differentiable function \(f\) such that \(A\) is the set of all non-differentiability points of \(f\). Since there are closed strongly symmetrically porous sets of Hausdorff dimension \(1\), our construction answers a problem posed by J. Foran in 1977. We also obtain results concerning smallness of the set of points at which a continuous function fulfills the symmetric Lipschitz condition but does not fulfill the ordinary Lipschitz condition.

We examine the existence of \(\kappa\)-to-1 functions \(f\colon\mathbb{R}\to\mathbb{R}\) in the class of continuous functions, Darboux functions, functions with perfect roads, and functions with the Cantor intermediate value property. In this setting \(\kappa\) denotes a cardinal number (finite or infinite). We also consider different variations on this theme.

We calculate the Hausdorff and packing dimensions of a generalized Cantor set which satisfies some condition on contracting ratios of its construction stages.

We generalize the theorem of Bernstein that any infinitely many times differentiable function on an interval, \(I\), that is regularly monotonic on \(I\) must be a real analytic function on \(I\).

The forcing relation was shown to be a partial order by S. Baldwin. Here we give a new, natural, geometric algorithm to determine if one cycle forces another.

In this paper we shall prove that a function \(f:[a,b] \to \overline{{\mathbb R}}\) that is \({\mathcal D}\)--integrable on \([a,b]\) can be defined as the limit of a \({\mathcal D}\)-controlled convergent sequence of stepfunctions (see the second part of Theorem 2). In the last section we show that Ridder’s \(\alpha\)- and \(\beta\)-integrals can also be defined as the limit of some controlled convergent sequences of stepfunctions (see Theorem 4).

The exact relations between the approximately continuous Perron and Denjoy integrals of Burkill [1] and Kubota [5, 6] are re-established by rectifying the faulty proofs of Kubota, and the related questions of Gordon [3] are resolved completely.

The maximal families (additive, multiplicative, lattice and with respect to the composition) for some special classes of strongly quasicontinuous functions are investigated.

If \(E_{\alpha,\beta}(x)\) is the generalized Mittag-Leffler function, then the complete monotonicity of \(E_{\alpha,\beta}(-x)\) for \(0\le\alpha\le1\), \(\beta\ge\alpha\) is an immediate corollary of a 1948 result due to Pollard. The proof can be accomplished within the framework of real analysis.

Jarnik in 1936 proved a remarkable property of directional cluster sets. This result states that for a function \(f\) defined on the open upper half plane to the extended real line, each pair of directional cluster sets intersect at all points on the real line but a countable set of points. An example was constructed to show that the exact analogue of Jarnik’s result fails for directional essential cluster sets. Here we shall establish a certain variant of this analogue for directional essential cluster sets of measurable functions.

The Cantor set is constructed by the iterate deletion of a middle interval equidistant from the end points. It is well known that the sums of points in the set cover completely the real line. It was an open problem to know if this property was still true for the sets obtained when the deleted interval is not any more equidistant from the end points. In this note we answer this question positively. We give a simple proof that reflects the geometric nature of the problem, and that is a variation on an old idea that goes back to Steinhaus.

In 1991, Agronsky and Cedar made the following conjecture: A continuum \(K \subset E^{k}\) is an orbit-enclosing \(\omega\)-limit set if and only if it is arcwise connected. The main aim of this paper is to disprove this conjecture by giving an example of an orbit-enclosing \(\omega\)-limit set \(S\) in \(E^{2}\) (cf. Theorem 3 below) which is not arcwise connected. Moreover, we show that \(S\) can be chosen with non-empty interior, and the mapping \(F\), with respect to which \(S\) is an orbit-enclosing \(\omega\)-limit set can be chosen as a triangular map.

Having defined for a couple of functions non co-supportivity in a given direction, we construct for any finite set of directions a couple \(f\) and \(g\) of possibly very regular, continuous except finite many points functions, which in any of these directions are non co-supportive and have the same indicatrix. Adding the requirement that both be continuous (on the joint interval of definition), we are still able to produce such a construction for two directions, but two seems to be the limit: it seems that such a result could not be had for three directions.

A natural problem is: what can we say about density points of sets with respect to measure \(\nu\), which has the same \(\sigma\)- ideal of null sets as another measure \(\mu\). We will show that, in general, the density points of \(A\) with respect to \(\mu\) need not be the density points of \(A\) with respect to \(\nu\), but we will find some sufficient conditions for \(\nu\) under which density points for \(\mu\) are also density points for \(\nu\).

We show some results connected with the problem whether it is consistent that every sup-measurable function \(F\colon\mathbb{R}^2\to\mathbb{R}\) is measurable. We will also relate this problem to a von Weizsäcker problem concerning a generalization of Blumberg’s theorem.

In this note, we shall prove that every absolutely Henstock integrable function is McShane integrable, without using the measurability of Henstock integrable functions and gauge functions.

We observe how a slight and even natural change in the Kurzweil-Henstock integral leads to a characterization of derivatives. We will also argue that the only way to characterize derivatives is by using some object or procedure which is at least as complicated as an integral.

We give a direct and simple proof of a result of Marczewski about graphs of \((s)\)-measurable functions. We also prove some related general results about graphs of measurable functions in metric spaces from which Marczewski’s result follows immediately.