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Just for fun, I was proving some results about convex functions the other day. I was able to show that for a convex set $E\subseteq\Bbb R,$ if $f:E\to\Bbb R$ is convex, then $f$ is left- and right-differentiable (and so continuous) on the interior of $E$ (though it needn't even be continuous at any boundary points contained in $E$). Moreover, if $x_0$ is an interior point of $E,$ then the left derivative of $f$ at $x_0$ is no greater than the right derivative of $f$ at $x_0.$

I am aware (though I haven't had a chance to try to prove) that the left and right derivative of such a function $f$ disagree at no more than countably-many points, but that got me wondering about a more general result.

Given an open convex subset $E\subseteq\Bbb R$ and a (not necessarily convex) function $f:E\to\Bbb R$ such that $f$ is left- and right- differentiable at every point of $E,$ can we conclude that $f$ fails to be differentiable at no more than countably-many points of $E,$ or do we need more information about $f$ to get there? What if we know that the right derivative dominates the left derivative everywhere? Is it possible that the right derivative strictly dominates the left derivative everywhere?

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  • $\begingroup$ Yes. Theorem 17.9 of Hewitt and Stromberg's Real and Abstract Analysis states: let $(a,b)$ be any open interval of $\Bbb R$ and let $f$ be an arbitrary real-valued function defined on $(a,b)$. Then there exist only countably many points $x\in(a,b)$ such that $f'_+(x)$ and $f'_{-}(x)$ both exist [they may be infinite] and are not equal. $\endgroup$ – David Mitra Jun 28 '15 at 18:55
  • $\begingroup$ Here's a link to the above. $\endgroup$ – David Mitra Jun 28 '15 at 19:05
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Yes. Theorem 17.9 of Hewitt and Stromberg's Real and Abstract Analysis states: let $(a,b)$ be any open interval of $\Bbb R$ and let $f$ be an arbitrary real-valued function defined on $(a,b)$. Then there exist only countably many points $x\in(a,b)$ such that $f′_+(x)$ and $f′_-(x)$ both exist [they may be infinite] and are not equal.

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