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Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ is convex on $\mathbb{R}$, but not necessarily differentiable everywhere. We know, though, that such a function will be differentiable almost everywhere, and that the set of points where $f$ is nondifferentiable, call this $\cal{A}$, is at most countable, and of Lebesgue measure zero.

Then, the derivative of $f$, call it $f'$, exists on $\mathbb{R}\setminus\cal{A}$.

Question #1: Is $f'$ continuous on $\mathbb{R}\setminus\cal{A}$?

Question #2: Does the result change if $f$ is nondecreasing?

I ask these questions because we know that a convex, differentiable function on $\mathbb{R}$ has a continuous derivative; I am just wondering if this proposition can be extended as described above.

Thanks!

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  • $\begingroup$ The statements and references at this 14 June 2001 ap-calculus post archived at Math Forum probably answer your questions, but I don't have time now to sort through this and be more specific. $\endgroup$ – Dave L. Renfro Nov 10 '17 at 12:34
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Yes to #1, no to #2. For any a.e. defined increasing function, left limits and right limits must exist, and they match except at jumps. $f'$ is an increasing function where defined, so is continuous except for jump discontinuities, which is where $f$ is not differentiable.

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