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

Problem: 

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

Answer: