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.
Best Answer
The existence of the limit $D^+\phi(x)$ tells us in particular that $\phi(x+h)\to\phi(x)$ as $h\to 0+$. Similarly, the existence of $D^-\phi(x)$ implies $\phi(x-h)\to\phi(x)$ as $h\to 0+$. These together yield that $\phi$ is continuous at $x$.