Intermediate value theorem

1. Aug 15, 2015

Suppose that $f$ is continuous on $[a,b]$ and let $M$ be any number between $f(a)$ and $f(b)$.
Then, there exists a number $c$ (at least one) such that:
$a < c < b$ and $f(c) = M$

Why did the author restrict $c$ to $(a,b)$ rather than $[a,b]$? After all, $c ∈ [a,b] ⇒ c ∈ (a,b)$.

2. Aug 15, 2015

micromass

False.

3. Aug 15, 2015

Oops. I just realized it's the other way around.
The question remains; why did the author restrict $c$ to $(a,b)$ rather than $[a,b]$?

4. Aug 15, 2015

micromass

5. Aug 16, 2015

FactChecker

You have to be careful here. If "M any number between f(a) and f(b)" includes f(a) and f(b), then you must include the end points a & b in the conclusion ( c is in [a,b] ). Otherwise, if M is properly between f(a) and f(b) but not equal to either, then you can conclude that c is in (a,b). And that is a stronger conclusion that c in [a,b].

6. Aug 20, 2015

Why is it a stronger conclusion? Is it because $c ∈ [a,b]$ implies more obvious conclusions like $c = a ⇒ f(c) = M = f(a)$?