What subsets of $\mathbb R$ can we speak of uniform continuity?

Question

From Ross's Elementary Analysis:

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The set $S$ can be an interval. But can it be a singleton? Can it contain isolated points such as $S = [2,4] \cup \{6\}$?

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Note: I'm not sure if this is relevant but this particular elementary real analysis book doesn't cover much of isolated points, limit points, open sets, closed sets, etc. I think it's in a chapter with an * (which I'm guessing means optional), but I notice that it avoids using terms such as interior, open set, etc unlike other books such as Lay - Analysis With An Introduction to Proof or Trench - Introduction to Real Analysis

Answer 1

To answer the question in the title:

What subsets of $\mathbb R$ can we speak of uniform continuity?

For any subset of $\mathbb{R}$. Reading definition 19.1 carefully, you would find that there is no restriction on $S$.

What confuses you might be in the second box. "It makes no sense to speak of a function being uniformly continuous at each point." What it really means is the following:

Suppose $S$ is a subset of $\mathbb{R}$ and $f:S\to\mathbb{R}$ be a real function. It makes no sense to speak of $f$ being uniformly continuous at some $x\in S$. It is not that $S$ cannot be a singleton.