Differentiability as continuity.

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.