In Signals & Systems: Second Edition by Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab, on page 362, it is stated that (with referenced figure shown below):
Signals at frequencies near [$\omega = 0$ and $\omega = 2\pi$] or any other even multiple of $\pi$ are slowly varying and therefore are all appropriately thought of as low-frequency signals. Similarly, the high frequencies in discrete time are the values of w near odd multiples of $\pi$. Thus, the signal $x_1[n]$ shown in Figure 5.3(a) with Fourier transform depicted in Figure 5.3(b) varies more slowly than the signal $x2[n]$ in Figure 5.3(c) whose transform is shown in Figure 5.3(d).
I am newly learning about the discrete-time Fourier transform, with (I think) a decent grasp on the continuous-time analog, which I started learning about a few weeks ago. I'm having more trouble with the discrete-time variant, though, and this quote is bothering me, because the book doesn't explicitly explain what it meant by this or why this is the case as far as I can find.
I understand that the DTFT of a signal $x[n]$, $X(e^{j\omega})$, is always periodic with period $2\pi$. Would the intuition behind this fact, then, be in the fact that odd multiples of $\pi$ are furthest away from the even multiples, and thus indicate quicker variation in the time-domain function, like how frequencies further from $0$ in a continuous-time FT indicate quicker variation in their time-domain function? This explanation seems to make sense to me, but I'm not sure I'm fully satisfied with it. Is there a more intuitive way to understand how the appearance of the DTFT of a function relates to the speed of variation of the $n$-domain function?
Because if your signal has a frequency of even multiple of $\pi$, you get $x_k=\sin(2k\pi+\phi)=\sin(\phi)$ which is constant.
With odd multiples of $\pi$ you get $x_k=\sin(k\pi+\phi)=(-1)^k\sin(\phi)$, which has likewise constant amplitude, but is alternating in sign.
If you take linear combinations with components of close-by frequencies, the sign behavior mostly stays the same, but the amplitude will become non-constant.