Does there exist continuous functios $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ \ $g\big((0,1)\big)$ ? The problem I am having is that since $(0,1)$ is not compact I am not able to tell anything decisive ; I know that the images of $f,g$ must be intervals but only that . What would happen if I replace $(0,1)$ by $[0,1)$ ? ( In general I am also interested in open bounded intervals $(a,b)$ and bounded half-open intervals $[a,b)$ ) . Please help . Thanks in advance .