Prove the Intermediate-Value Theorem

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Lesson Parent: 
The Intermediate-Value Theorem
Problem: 

Prove the Intermediate-Value Theorem. We assume that $f(x)$ is continuous on $[a,b]$ and that $f(a)\neq f(b)$. Prove that for every number $\Omega$ between $f(a)$ and $f(b)$ that there is at least one number $\omega$ in $(a,b)$ such that $f(\omega)=\Omega$.

SAMPLE SOLUTION

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Solution: 

We define <Sign in to see all the formulas> as an arbitrary value between $f(a)$ and $f(b)$. When we say that <Sign in to see all the formulas> is between $f(a)$ and $f(b)$ we can have either <Sign in to see all the formulas> or <Sign in to see all the formulas>. This means we have to prove the intermediate-value theorem for both of these scenarios.

Let's define <Sign in to see all the formulas>. Since $f(x)$ is continuous on $[a,b]$, we know that $e(x)$ is continuous on $[a,b]$.

Now when <Sign in to see all the formulas> we can subtract <Sign in to see all the formulas> to see that <Sign in to see all the formulas>. When <Sign in to see all the formulas> we can subtract <Sign in to see all the formulas> to see that <Sign in to see all the formulas>.

Since $e(x)$ is continuous on $[a,b]$ and that <Sign in to see all the formulas> or <Sign in to see all the formulas> then there is a number <Sign in to see all the formulas> between $a$ and $b$ such that <Sign in to see all the formulas>. Since <Sign in to see all the formulas> then <Sign in to see all the formulas>.