Does the second derivative of a convex function exist at all but countably many points?

Question

I know that the second derivative of a convex function exists almost everywhere (Alexandrov's theorem).

And I know that the first derivative exists everywhere except countably many points (This question).

So, does the second derivative exist everywhere except countably many points? I'd appreciate a source I can cite, or a counterexample.

Answer 1

If $g(t)$ is a nondecreasing real-valued function, then $f(x)=\int_a^x g(t)\,dt$ is convex. If $g$ is also continuous, then $f'(x)=g(x)$ everywhere. If $g(t)$ is the Cantor function, then $g$ is nondifferentiable at uncountably many points.

Gap: I do not have a reference or proof handy for showing that $g$ is nondifferentiable at uncountably many points in the Cantor set.

For a reference on the nondifferentiability set of the Cantor function, see Darst's "The Hausdorff Dimension of the Nondifferentiability Set of the Cantor Function is $[\ln(2)/\ln(3)]^2$" (JSTOR link), which I found in the references on the Wikipedia page linked above.