Prove that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$ or $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

Prove that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$ or $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.

Answer: 

It is true that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$ or $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.

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Solution: 

We have proven both cases here and here.