Continuous bijection from (0,1)(0,1) to [0,1][0,1]

Does there exist a continuous bijection from (0,1) to [0,1]? Of course the map should not be a proper map.

Answer

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 [xε,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.

Attribution
Source : Link , Question Author : Alex , Answer Author : t.b.

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