I was reading the proof of intermediate value theorem: https://en.wikipedia.org/wiki/Intermediate_value_theorem#Proof
But I cannot understand the last sentence of the statement below.
Define $S = \lbrace x \in [a, b] \; | \; f(x) \leq u \rbrace$ and $c$ as the supremum of $S$. Choose some $\epsilon$. By continuity, there exists $\delta$ such that $|f(x) - f(c)| < \epsilon$. By the properties of the supremum, we can chose some $a^*$ from $(c - \delta, c]$ such that $a^* \in S$.
Suppose that the function looks like below and we choose $u$ as indicated on the axis. The set $S$ is colored in blue. In this case, $c$ is an isolated point and thus, we cannot guarantee the existence of $a^*$ within $\delta$-neighborhood of $c$.
Clearly I have misunderstood about something, but where did I get wrong?
Edit: In the above case, $a^* = c$, but now I'm wondering how we can guarantee that $c = \sup S \in S$.