# 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: 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?

Tags :