### Differentiability as continuity.

David Gauld and Frédéric Mynard
Source: Real Anal. Exchange Volume 31, Number 2 (2005), 425-430.

#### 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.

First Page:
Primary Subjects: 26A24, 54C30
Secondary Subjects: 26A06, 26A27, 54A10, 54A20
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.rae/1184104035
Zentralblatt MATH identifier: 1146.26304
Mathematical Reviews number (MathSciNet): MR2265784

### References

E. Kronheimer, R. Geroch and G. McCarty, No topologies characterize differentiability as continuity, Proc. Amer. Math. Soc., 28 (1971), 273–274.
Mathematical Reviews (MathSciNet): MR271969
Zentralblatt MATH: 0196.27004
Digital Object Identifier: doi:10.1090/S0002-9939-1971-0271969-5
A. Machado, Quasi-variétés complexes, Cahiers Top. Géom. Diff., 11 (1969), 229–279.
Mathematical Reviews (MathSciNet): MR276494