Answer
The Intermediate Value Theorem:
Suppose that
- $f$ is continuous on the closed interval $[a, b]$
- let $N$ be any number between $f(a)$ and $f(b)$, $f(a)\ne f(b)$
then there exists a number $c$ in $(a,b)$ such that $f(c)=N$
Work Step by Step
The Intermediate Value Theorem:
Suppose that
- $f$ is continuous on the closed interval $[a, b]$
- let $N$ be any number between $f(a)$ and $f(b)$, $f(a)\ne f(b)$
then there exists a number $c$ in $(a,b)$ such that $f(c)=N$
You can check the theorem in the Continuity section.
This theorem can be intuitively thought as true as follows:
- N is a number between $f(a)$ and $f(b)$
- We draw a horizontal line $y=N$
- If the graph has no break, then it must intersect the horizontal line $y=N$ at a point $c$ between $a$ and $b$.
