For a given function in some classes related to real derivatives, we examine the structure of the set of points which are not Lebesgue points. In particular, we prove that for a summable approximately continuous function, the non-Lebesgue set is a nowhere dense nullset of at most Borel class 4.

We present a method of integration along the lines of the Henstock-Kurzweil integral. All $L^r$-derivatives are integrable in this method.

A function $f:ℝ\to ℝ$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers ${\alpha }_0,...,{\alpha }_{n-1}$ such that $s^{n+1}{\int }_0^{\delta }{e}^{-st}[f(x+t)-{\sum }_{i=0}^{n-1}{\alpha }_i{t}^i/i!]dt$ converges as $s\to +\infty$ for some $\delta >0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_{\langle n\rangle }\left(x\right)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than the generalized...

Let $fH\subset ℝ\to ℝ$ be $k$ times differentiable in both the usual (iterative) and Peano senses. We investigate when the usual derivatives and the corresponding Peano derivatives are different and the nature of the set where they are different.