an:05241679
Zbl 1164.26007
Broszka, Dorota; Grande, Zbigniew
On \(\mathcal I\)-differentiation
EN
Tatra Mt. Math. Publ. 35, 25-40 (2007).
00216632
2007
j
26A24
\({\mathcal I}\)-derivative; direction; strongly \({\mathcal I}\)-derivative
Let \({\mathcal K}\) be a field of all reals (or of all complex numbers) and let \({\mathcal I}\subset {\mathcal K}\) be a set such that 0 is an accumulation point of \, \(\mathcal I\). Let \(Y\) be a real (or a complex) Banach space. If \(f\:{\mathcal K} \to Y\) is a function then \(D^{{\mathcal I}}_h f(x_0)=\lim \limits _{{\mathcal I}\ni r\to 0} \frac {f(x_0+rh)-f(x_0)}{r}\) is called the \({\mathcal I}\)-derivative of \(f\) at the point \(x_0\in {\mathcal K}\) in the direction \(h\in {\mathcal K}\) when ever this limit exists and belongs to \(Y\). If \({\mathcal K}={\mathcal I}\) and \(h=1\) then we obtain the ordinary derivative of \(f\) at the point \(x_0\). The limit \(Ds^{{\mathcal I}}_h f(x_0)=\lim \limits _{{\mathcal K}\times {\mathcal I}\ni (x,r)\to (x_0, 0)} \frac {f(x+rh)-f(x)}{r}\) is called strong \({\mathcal I}\)-derivative of \(f\) at \(x_0\in {\mathcal K}\) in the direction \(h\in {\mathcal K}\). Some properties of \({\mathcal I}\)- derivatives and strong \({\mathcal I}\)-derivatives are investigated.
J??n Bors??k (Ko??ice)