I have a general question.
Say $f$ is a real function from $(a,b)$ to $\mathbb R$. We usually prove continuity on $[a,b]$, but if f were continuous on the open interval $(a,b)$, would there be any issues with differentiability? Do we need continuity on $[a,b]$ in order to have differentiability on (a,b)?
No. Let's start from the claim that says that differentiability at a point implies continuity at that point. Differentiability on an interval $(a,b)$ is equivalent with differentiability at each point of that interval. This implies that the function is continuous at each point of that interval, i.e. continuous on the interval $(a,b)$. This says nothing about the endpoints.
As an example, consider the function $\tan(x)$. It is continuous on $(\pi/2,\pi/2)$, but not on $[-\pi/2,\pi/2]$ (it isn't even defined at the endpoints). Still, it is differentiable on $(-\pi/2, \pi/2)$.