Porosity and several different variations of porosity have been used to describe small sets for some time. A common link between these notions is that off of the real line these small sets are not necessarily disconnected. In this paper, shell porosity is introduced and some properties of shell porous sets are investigated. This includes the fact that in a complete metric space, any closed set which may be expressed as a countable union of shell porous sets must be totally disconnected.

The paper contains comparisons of classes functions whose restrictions to some special sets of positive outer measure have continuity points, quasi-con tin uity points or are cliquish at some points.

We apply a type I metric outer measure construction to give a further new proof for the existence of open-invariant measures on a compact metric spaces.

In this note we provide another direct proof of the Lebesgue integrability of McShane integrable functions

For topological spaces $X$, $Y$ let $Y^X$, $C(X,Y)$ denote the sets of all functions from $X$ to $Y$ and of all continuous functions from $X$ to $Y$ respectively. We consider nets of functions from these spaces which are evenly continuous or pointwise equicontinuous, and for such nets the relationship between pointwise convergence and topological convergence is studied. We find that in $Y^X$ for an evenly continuous net pointwise convergence (to a function from $Y^X$) implies topological convergence. Conversely, if $Y$ is an uniform space and $(f_i)$ a pointwise equicontinuous net then $\mathrm{\Gamma }{f}_1$ lim inf $\mathrm{\Gamma }{f}_i$ implies the pointwise convergence of $(f_i)$ to $f$. By an example is shown that in the second assertion the equicontinuity of $(f_i)$ cannot be replaced by even continuity. As corollaries of our results we get some results of T. Neubrunn and Ľ. Holà [6] and of G. Beer [1] respectively. Moreover, a relationship to the lower semi-finite graph-topology is established.

In this paper we introduce the property of the arcwise almost continuity and investigate relationships between the notions of almost continuity and arcwise almost continuity.

In this paper a generalization of a theorem of Bor [2] has been proved.

In this paper, the authors obtain several characterizations of $\beta$-continuity due to Abd El-Monsef et al. [1] and show that almost quasi-continuity in the sense of Borsik and Doboš [3] is equivalent to $\beta$-continuity.

In this paper it is shown that every closed graph function is locally ${\text{w}}^{*}$ continuous, if the range space is locally compact and Hausdorff. Also presented is a decomposition of continuity into separate continuity and local ${\text{w}}^{*}$ continuity, thus generalizing a result involving theclosed graph property. A decomposition of continuity into connectedness and local ${\text{w}}^{*}$ continuity is given for functions whose domain spaceis locally connected.

The main purpose of the present note is to make a first step in stating L’Hôpital’s rule for functions of a real variable with higher-dimensional ranges. This is done by working with absolutely continuous functions in the framework of Banach modules.

In this paper we show that under Martin’s Axiom there is a Hamel basis $H$ for which the set $E^{+}(H)$ of all finite linear combinations of elements from $H$ with nonnegative rational coefficients is both of Lebesgue measure zero and of the first Baire category. We also obtain that the additive group of the reals $ℝ$ is a direct sum of two subgroups one of which is uncountable, of measure zero and of the first Baire category.

It is shown that the set of points at which a real valued function of a real variable is qualitatively continuous and finitely qualitatively symmetrically differentiable but not qualitatively differentiable is a $\sigma$-symmetrically porous set.

We charaterize the set where an almost everywhere continuous Baire $1$ function is not a.e. continuous in the O’Malley sense.

It is known that the family of all derivatives (from $ℝ$ into $ℝ$) whose product with every continuous function is a derivative is the same as the family of all locally summable derivatives such that

for each $x\in ℝ$. In this paper we prove an analogous theorem in multidimensional case.

In this paper we prove that each Baire one function $u:{ℝ}^m\to ℝ$ can be written as $u=f\cdot g+h$, where $f,g,h:{ℝ}^m\to ℝ$ are non-degenerate derivatives (both notions with respect to the ordinary differentiation basis).