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On \(\mathcal I\)-differentiation. (English) Zbl 1164.26007
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.
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems