7.2: Proof of the Intermediate Value Theorem

Learning Objectives

• Proof of Intermediate value theorem

We now have all of the tools to prove the Intermediate Value Theorem (IVT).

Theorem $\PageIndex{1}$: Intermediate Value Theorem

Suppose $f(x)$ is continuous on $[a,b]$ and v is any real number between $f(a)$ and $f(b)$. Then there exists a real number $c ∈ [a,b]$ such that $f(c) = v$.

Sketch of Proof

We have two cases to consider: $f(a) ≤ v ≤ f(b)$ and $f(a) ≥ v ≥ f(b)$.

We will look at the case $f(a) ≤ v ≤ f(b)$. Let $x_1 = a$ and $y_1 = b$, so we have $x_1 ≤ y_1$ and $f(x_1) ≤ v ≤ f(y_1)$. Let $m_1$ be the midpoint of $[x_1,y_1]$ and notice that we have either $f(m_1) ≤ v$ or $f(m_1) ≥ v$. If $f(m_1) ≤ v$, then we relabel $x_2 = m_1$ and $y_2 = y_1$. If $f(m_1) ≥ v$, then we relabel $x_2 = x_1$ and $y_2 = m_1$. In either case, we end up with $x_1 ≤ x_2 ≤ y_2 ≤ y_1,\; y_2 - x_2 = \frac{1}{2} (y_1 - x_1)$, $f(x_1) ≤ v ≤ f(y_1)$, and $f(x_2) ≤ v ≤ f(y_2)$.

Now play the same game with the interval $[x_2,y_2]$. If we keep playing this game, we will generate two sequences ($x_n$) and ($y_n$), satisfying all of the conditions of the nested interval property. These sequences will also satisfy the following extra property: $∀ n,\; f(x_n) ≤ v ≤ f(y_n)$. By the NIP, there exists a $c$ such that $x_n ≤ c ≤ y_n,\; ∀ n$. This should be the $c$ that we seek though this is not obvious. Specifically, we need to show that $f(c) = v$. This should be where the continuity of $f$ at $c$ and the extra property on ($x_n$)and ($y_n$) come into play.

Exercise $\PageIndex{1}$

Turn the ideas of the previous paragraphs into a formal proof of the IVT for the case $f(a) ≤ v ≤ f(b)$.

Exercise $\PageIndex{2}$

We can modify the proof of the case $f(a) ≤ v ≤ f(b)$ into a proof of the IVT for the case $f(a) ≥ v ≥ f(b)$. However, there is a sneakier way to prove this case by applying the IVT to the function $-f$. Do this to prove the IVT for the case $f(a) ≥ v ≥ f(b)$.

Exercise $\PageIndex{3}$

Use the IVT to prove that any polynomial of odd degree must have a real root.