If a function $f$ is continuous at every point $a$ in an interval $I$, we'll say that $f$ is continuous on $I$.
The Intermediate Value Theorem talks about the values that a continuous function has to take:
| Theorem: Suppose $f(x)$ is a continuous function on the interval $[a,b]$ with $f(a) \ne f(b)$. If $N$ is a number between $f(a)$ and $f(b)$, then there is a point $c$ between $a$ and $b$ such that $f(c)=N$. |
In other words, to go continuously from $f(a)$ to $f(b)$, you have to pass through $N$ along the way. In this video we consider the theorem graphically and ask: What does it do for us?
We can use the Intermediate Value Theorem (IVT) to show that certain equations have solutions, or that certain polynomials have roots. For instance, the polynomial $f(x)=x^4+x-3$ is complicated, and finding its roots is very complicated. However, it's easy to check that $f(-2) = 11$ and $f(0)=-3$ and $f(2)=15$. Since $11 \gt 0 \gt -3$, there has to be a point $c$ between $-2$ and $0$ with $f(c)=0$. Likewise, since $-3 \lt 0 \lt 15$, there has to be a point $c'$ between $0$ and $2$ with $f(c')=0$. In other words, $f(x)$ has a root somewhere between $-2$ and $0$ and another root between $0$ and $2$. We don't know exactly where these roots are, but we know they exist.