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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 \(i= 0, 1,\dots, k\).

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