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An example of construction of pseudo-numbers by methods of recursion theory. (Russian) Zbl 0728.03033
The paper under review is closely related to the (pseudo) differentiability of constructive functions of a real variable, thoroughly investigated by the author in his previous papers. A survey of the results can be found a joint paper of the author and P. Filipec [Math. logic and its applications, Proc. Adv. Int. Summer Sch. Conf., Druzhba 1986, 81-106 (1987; Zbl 0699.03034)].
Let A be the set of all arithmetic real numbers from the unit interval. Let \(D_{kl}(+\infty,G,X)\) denote “the pseudo-derivative of a constructive function (CF) G at \(X\in A\) equals \(+\infty ''\). If F is a CF satisfying some additional conditions, then there is a monotone CF G such that \(\neg D_{kl}(+\infty,G,X)\) implies either the pseudo- differentiability of F at X, or that the lower semi-derivative of F at X equals -\(\infty\) and the upper one equals \(+\infty\) (for all \(X\in A)\). For details cf. the author’s paper in Commentat. Math. Univ. Carol. 21, 457-472 (1980; Zbl 0485.03034). Thus the set \(A_{\vartheta}\) of all \(X\in A\) such that there is no CF G satisfying \(D_{kl}(+\infty,G,X)\) is of interest. The paper under review is devoted to the investigation of \(A_{\vartheta}\) and of some other sets related to \(A_{\vartheta}\).
03F60 Constructive and recursive analysis
26E40 Constructive real analysis
47S30 Constructive operator theory