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Differentiability as continuity. (English) Zbl 1146.26304

Summary: We characterize differentiability of a map \(f:\mathbb R\rightarrow \mathbb R\) in terms of continuity of a canonically associated map \(\widehat{f}\). To characterize pointwise differentiability of \(f,\) both the domain and range of \(\widehat{f}\) can be made topological. However, the global differentiability of \(f\) is characterized by the continuity of \(\widehat{f}\) whose domain is topological but whose range is a convergence space.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A06 One-variable calculus
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
54C30 Real-valued functions in general topology
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References:

[1] E. Kronheimer, R. Geroch and G. McCarty, No topologies characterize differentiability as continuity , Proc. Amer. Math. Soc., 28 (1971), 273-274. · Zbl 0196.27004 · doi:10.2307/2037799
[2] A. Machado, Quasi-variétés complexes , Cahiers Top. Géom. Diff., 11 (1969), 229-279. · Zbl 0199.59003
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