Intermediate value theorem

[PFuser1232]

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, $c ∈ [a,b] ⇒ c ∈ (a,b)$.

 

[micromass]

After all, $c ∈ [a,b] ⇒ c ∈ (a,b)$.

False.

 

[PFuser1232]

False.

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

 

[micromass]

Because it gives more information.

 

[FactChecker]

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, $c ∈ [a,b] ⇒ c ∈ (a,b)$.

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].

 

[PFuser1232]

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].

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

 

[HallsofIvy]

It is a stronger conclusion because it restricts the region in which "c" lies- it gives more information. Using "between" to mean "strictly between" is a stronger statement.