Intermediate value theorem wikipedia proof I have stubled upon the following proof on Wikipedia. I wonder if it is right.
First I think that S must be open interval, right? That is not explicit. Second what troubles me is the a* definition I think it should be a* interval should be open on c so that $\upmu-\epsilon$ would be higher than $f(a*) -\epsilon$, since \upmu never reaches $f(b).

The set $S$ is not necessarily an interval, but it is open, which is what is needed for the proof; this follows from the continuity of $f$. 
It is true that $a^*$ can never be equal to $c$ (because of $S$ being open). But the interval doesn't play any role. All that you need is that $c-\delta<a^*<c$. 
