No topologies characterize differentiability as continuity

Abstract:

Do there exist topologies $\mathcal {U}$ and $\mathcal {V}$ for the set $R$ of real numbers such that a function $f$ from $R$ to $R$ is smooth in some specified sense (e.g., differentiable, ${C^n}$, or ${C^\infty }$) with respect to the usual structure of the real line if and only if $f$ is continuous from $\mathcal {U}$ to $\mathcal {V}$? We show that the answer is no.