I understand that neither $\tan(t)$ nor its derivative $\sec^2(t)$ are defined at $t = k\pi /2$, where $k$ is any integer. If I plug in $x(t) = \tan(t)$, that leaves me with $\sec^2(t) = 1 + \tan^2(t)$ and I see no problem with that (trig identity). Both $x'$ and $x$ have the same domain, so I don't see why I'd have to limit it to $(-\pi/2, \pi/2)$. I read somewhere that the solution has to be smooth? I understand that $x(t)$ has to be differentiable, because we're using its derivative in the DE. My question is why does the solution have to be one continuous line? Why can't it be two or more continuous lines? What's so bad about discontinuities if they line up?
P.S. The problem I'm referring to is Example 7.7 found on page 78 in Differential Equations by Polking, Boggess, and Arnold 2nd Edition.