Continuous versus Discrete Spectrum in Ergodic Theory

Question

Inside Ergodic Theory, we can find these definitions:

Let $f:X\rightarrow X$ measurable and $(X,\mathcal{B},\mu)$ measure space, and $U_f$ the Koopman Operator.

1. A pair $(f,\mu)$ is said to have discrete spectrum if the proper vectors of $U_f$ are a basis for $L^2(X,\mu)$.

2. A pair $(f,\mu)$ is said to have continuous spectrum if 1 is the only proper value, and the only proper vectors of $U_f$ are constant functions.

My question is: why do they have these names? Are they, in some sense, complementary to each other?

I mean, what is the reason for this dichotomy, if it ever exists?