Real Analysis Exchange

Differentiability as continuity.

David Gauld and Frédéric Mynard

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Abstract

We characterize differentiability of a map $f:\mathbb{R\rightarrow R}$ 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.

Article information

Source
Real Anal. Exchange, Volume 31, Number 2 (2005), 425-430.

Dates
First available in Project Euclid: 10 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.rae/1184104035

Mathematical Reviews number (MathSciNet)
MR2265784

Zentralblatt MATH identifier
1146.26304

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 54C30: Real-valued functions [See also 26-XX]
Secondary: 26A06: One-variable calculus 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)

Keywords
real valued functions differentiability continuity convergence spaces

Citation

Gauld, David; Mynard, Frédéric. Differentiability as continuity. Real Anal. Exchange 31 (2005), no. 2, 425--430. https://projecteuclid.org/euclid.rae/1184104035


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References

  • E. Kronheimer, R. Geroch and G. McCarty, No topologies characterize differentiability as continuity, Proc. Amer. Math. Soc., 28 (1971), 273–274.
  • A. Machado, Quasi-variétés complexes, Cahiers Top. Géom. Diff., 11 (1969), 229–279.