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?
Best Answer
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.