Let f be a measurable function such that ${\Delta }_k(x,h;f)={O\left(\right|h|}^{\lambda })$ at each point x of a set E, where k is a positive integer, λ > 0 and ${\Delta }_k(x,h;f)$ is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative $f_{\left(k\right)}$ exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative $f_{\left(\right[\lambda \left]\right)}$ exists a.e. on E and that the result is false if λ = k-1; it is further proved that if λ is any positive integer and if the approximate Peano derivative...

We show that any quasi-arithmetic mean $A_{\phi }$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f\left(M(x,y)\right)=A_{\phi }(f\left(x\right),f\left(y\right))$ and $f\left(A_{\phi }(x,y)\right)=M(f\left(x\right),f\left(y\right))$ are the constant ones.

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. $L^p$, 1 ≤ p ≤ ∞) sense at $x\in {ℝ}^m$ if there are numbers $f_{\alpha }\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum }_{\left|\alpha \right|\le n}{f}_{\alpha }\left(x\right)h^{\alpha }/\alpha !$ is $O\left(h^u\right)$ in the approximate (resp. $L^p$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or $L^p$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and f = g on Π....

Automatic differentiation is an effective method for evaluating derivatives of function, which is defined by a formula or a program. Program for evaluating of value of function is by automatic differentiation modified to program, which also evaluates values of derivatives. Computed values are exact up to computer precision and their evaluation is very quick. In this article, we describe a program realization of automatic differentiation. This implementation is prepared in the system UFO, but its...