Let $f:[a,b]\mapsto[a,b]$ be continous, suppose $f(a)=f(b)$. Show that there exist $c \in [a,\frac{b+a}{2}]$ with $f(c)=f(c+\frac{b-a}{2})$
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$