I am reading "Introduction to the Theory of Computation" (2ed) by Michael Sipser. In Example 1.35, it says the NFA accepts $\epsilon$, which I understand, since the state can stay at $q_1$ upon input of $\epsilon$:
But in Example 1.38, it shows to be an empty set for $q_1$, $q_3$, and $q_4$:
I thought it should be $\delta(q_1, \epsilon)=\{q_1\}$ at least, for instance. I always assume there is an "implicit self-$\epsilon$-transition" at every state, but why it is $\emptyset$ now? I also noticed the lack of a "self-$\epsilon$-transition" in the proof of Theorem 1.45 (no staying at $q_0$ upon the input of $\epsilon$). Have I missed something important?