Questions about proof of intermediate value theorem

by Dimen   Last Updated June 30, 2020 02:20 AM

I was reading the proof of intermediate value theorem:
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 $c$.

enter image description here

Clearly I have misunderstood about something, but where did I get wrong?

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