This is an excerpt from a textbook on quantum mechanics. $u(x)$ is a square-integrable complex-valued function of a real variable.
The assumption that $u(x)\neq 0$ is probably there, too. It seems to imply that these two statements are equivalent:
- $u(x)$ and $u'(x)$ are continuous at $x=a$.
- $\frac{u'(x)}{u(x)}$ is continuous at $x=a$.
It is easy to prove that the former implies the latter, but what about the other way around? Is it true? If so, how to prove it? If not, could there be an assumption that I've missed, or have I misinterpreted the text?