I define a continuity set $C$ to be a subset of $\mathbb{R}$ such that there exists a real function $f$ whose set of real numbers $r$ such that $f$ is continuous at $r$ is precisely the set $C$. Similarly, I define a differentiability set $D$ to be a subset of $\mathbb{R}$ such that there exists a real function $f$ whose set of real numbers $r$ such that $f$ is differentiable at $r$ is precisely the set $D$. It is an interesting fact that a set is a continuity set precisely when it a $F_\sigma$ set. Thus, there is a topological characterization of continuity sets. I wonder, is there a, perhaps more complicated, topological characterization of differentiability sets? Or is the answer unknown at this time?
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1The notion of continuity set can be defined on any metric space, but differentiability is not a topological property. That doesn't give an answer, but it does indicate it might be negative. Have you found a continuity set that can be proven not to be a differentiability set? It seems the most obvious approach might be to prove that a continuity set is a differentiability set, and visa versa. But that will only work if it is true. :) – Thomas Andrews Jun 17 '25 at 21:55