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!
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.