Does there exist a continuous bijection from (0,1) to [0,1]? Of course the map should not be a proper map.
No. If f:(0,1)→[0,1] were continuous and bijective, there would be a unique point x∈(0,1) such that f(x)=1. However, since f is continuous, the intervals [x−ε,x] and [x,x+ε] would be mapped to intervals [a,1] and [b,1], say. By bijectivity we’d have a,b<1. Thus every value strictly between max and 1 would be assumed at least twice, contradicting bijectivity.