Intermediate value theorem confusion for decreasing function? I am told by my teacher that the IVT states the following:
Let f be continuous on the closed and bounded interval [a, b], and let f(a) ≤ y ≤ f(b). Then there exists a value c on [a, b] such that f (c) = y.
However, I'm confused at why f(b) is assumed to be larger than f(a). Certainly, b is larger than a, but if the function is, say, strictly decreasing...
Basically, I'm confused about the definition of the IVT.
 A: If $f(b) \leq y \leq f(a)$ then also there exist $c$ such that $f(c)=y$. There is not much difference between these two cases since changing $f$ to $-f$ and $y$ to $-y$ will yield this new result. Just for definiteness they have assumed that $f(a) \leq y \leq f(b)$. 
A: As you wrote it, the theorem is stated for one case. It is just cumbersome to write it in general. And you can easily get the other case by applying the theorem to $-f(x)$. 
A: Basically, there are 3 correct statements


*

*Let $f$ be continuous on the closed and bounded interval $[a, b]$, and $f(a) \le f(b)$. Let $f(a) ≤ y ≤ f(b)$. Then there exists a value $c$ on $[a, b]$ such that $f (c) = y$.

*Let $f$ be continuous on the closed and bounded interval $[a, b]$, and $f(a) \ge f(b)$. Let $f(a) \ge y \ge f(b)$. Then there exists a value $c$ on $[a, b]$ such that $f (c) = y$.

*Let $f$ be continuous on the closed and bounded interval $[a, b]$, Let $y$ be between $f(a)$ and $f(b)$. Then there exists a value $c$ on $[a, b]$ such that $f (c) = y$. (But note that it requires to define word "between" beforehand (because it may mean different things) or to rewrite it using cases)
It's easy to see (But please try to understand that formally) they trivially follow one from another. (3=> 1, 3=>2 just special cases, 1<=>2 by replacing $f$ with $-f$, (1and 2)=>3 because either $f(a)\le f(b)$ or $f(a) \le f(b)$ (or both))
Normally you would understand IVT to be 3rd, but it's reasonable to want to prove only 1st (with understanding that it's general enough) then 3rd because you don't need to explicitly work with two cases (and also don't need to define term "between" which is not always done)
