Question 9
(NET) Let $f(x)$ be a continuous function defined on the closed interval $0\le x\le1$. Let $G(f)$ denote the graph of $f(x)$: $G(f)=\{(x,y)\in\mathbb R^2\mid0\le x\le1,y=f(x)\}$. Let $G_a(f)$ denote the graph of the translated function $f(x-a)$ (translated over a distance $a$), defined by $G_a(f)=\{(x,y)\in\mathbb R^2\mid a\le x\le a+1,y=f(x-a)\}$. Is it possible to find for every $a$, $0\lt a\lt 1$, a continuous function $f(x)$, defined on $0\le x\le1$, such that $f(0)=f(1)=0$ and $G(f)$ and $G_a(f)$ are disjoint point sets?