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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.
##### MSC:
 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems