If $\phi$ is finite and convex in $(a, b)$, then $\phi$ is continuous in $(a, b)$. Moreover, $\phi'$ exists except at most in a countable set and is monotone increasing.
Theorem 2.8 Every function of bounded variation has at most a countable number of discontinuities, and they are all of the first kind (jump or removable discontinuities).
- How does $(7.41)$ show in particular that $\phi$ is continuous in $(a,b)$? Don't we need to show $D^-\phi(x) = D^+\phi(x)$ as $h\to 0+$? The book shows $D^-\phi(x) = D^+\phi(x)$ at the end of the proof.