real analysis - Does there exists a onto continuos function $fcolon (0,1) to[0,1]$ which is one to one .

Iam trying to solve this problem , i can visualize graphically it is not possible. iam trying to do the proof by contradiction. if f:(0,1) to [0,1] is a continuos onto function then i have to prove that f can never be 1-1 . so iam letting that if f is 1-1 then either f is strictly increasing or strictly decreasing function. so case (1) if f is strictly increasing function then there exits some t in (0,1) such that f(t)=0 , so for x lesser than t f(x)k , f(x) <0 which is also not possible .so f can't be 1-1 . My method is right ??? is there is any alternative approach then plz tell