Can someone explain to me why we can conclude that the angle is of rotation is the same? I understand everything but the part where it says it depends continuously and the remaining portion. I don’t get where that comes from. Why is it equal to $a$ and not $-a$? Why can only one value occur?
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The fact that both $M'$ and $B$ depend continuously on $\beta$ implies that $\alpha'$ is a continuous function of $\beta$, that is $\lim_{\beta\to\beta_0} \alpha'(\beta) = \alpha'(\beta_0)$. Now it is obvious that for $\beta = 0$ we must have $M' = M$, and consequently
$$\alpha = \alpha'(0) = \lim_{\beta\to 0} \alpha'(\beta) = \lim_{\beta\to 0} \pm \alpha = \pm \alpha \quad\quad(*)$$
where the last equality follows from the fact that $\alpha$ is independent of $\beta$. It is then mandatory that $\alpha' = \alpha$, for otherwise $(*)$ would be a contradiction for any $\alpha \neq 0$.
