Finding x and y for continuous onto function

Question

$f:[0,1]\rightarrow [0,1]$ continuous onto function such that $f(0)=0=f(1)$. Then show that there exists distinct values $x,y$ belongs to $[0,1]$ such that $f(x)=\frac{1}{2}=f(y)$. could you please tell some hint?

Answer 1

Since $f$ is surjective there is a point $a \in [0,1]$ such that $f(a)=1$. Then we have $f(0)=0$ and $f(a)=1$ so by continuity of $f$ there is $x \in (0,a)$ such that $f(x)=1/2$. Similarly $f(a)=1$ and $f(1)=0$ so by continuity of $f$ there is $y\in(a,1)$ such that $f(y)=1/2$. It cannot be that $x=y$ since $x<a<y$ and so the claim is proved.