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Extending Peano derivatives. (English) Zbl 0824.26003
Let $$H\subset [0, 1]$$ be a closed set and let $$f: H\to \mathbb{R}$$. Then $$f$$ is $$k$$-times Peano differentiable at $$x\in H$$ relative to $$H$$ if there are numbers $$f_ 1(x, H),\dots, f_ k(x, H)$$ such that $$y\in H$$ implies that $f(y)= f(x)+ (y- x) f_ 1(x, H)+\cdots+ \textstyle{{(y- x)^ k\over k!}} (f_ k(x, h)+ \varepsilon_ k(y)),$ where $$\lim_{y\in H, y\to x} \varepsilon_ k(y)= 0$$. The number $$f_ k(x, H)$$ is called the $$k$$th Peano derivative of $$f$$ at $$x$$ relative to $$H$$.
If $$k\geq 0$$ is an integer, $$H$$ is a perfect set of finite Denjoy index and $$f_ k(x, H)$$ exists for every $$x\in H$$, then there exists a $$k$$- times Peano differentiable function $$F: [0, 1]\to \mathbb{R}$$ such that $$F_ i(x)= f_ i(x, H)$$ for every $$x\in H$$ and $$i= 0, 1,\dots, k$$.

##### MSC:
 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
##### Keywords:
Peano derivatives; Denjoy index
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