2.5 The Intermediate Value Theorem
Notes
The Intermediate Value Theorem (IVT): if $f$ is continuous on $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there is at least one $c$ in $[a, b]$ with $f(c) = N$.
- Continuity on the whole closed interval is required.
- It guarantees that a value exists, but does not say where, or how many.
- Example: $f$ continuous on $[1, 4]$ with $f(1) = -2$ and $f(4) = 6$ must take the value 3 somewhere.
Practice
Which condition is required to use the IVT on $[a, b]$?
The IVT only needs continuity on the closed interval.
$f$ is continuous on $[1, 4]$, with $f(1) = -2$ and $f(4) = 6$. For which value of $N$ does the IVT NOT guarantee a $c$ with $f(c) = N$?
Only values between $-2$ and 6 are guaranteed. The number 8 is outside that range.
Why can't the IVT be used to show $\displaystyle f(x) = \frac{1}{x}$ equals 0 somewhere on $[-1, 1]$?
$f$ is undefined at 0, so it is not continuous on the whole interval.
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