Question

Prove that Continuous image of a connected set is Connected.​

Answer 1

Answer:

In the proof, we started with consider f(E)=A∪B, where A and B are nonempty separated subsets. Then put G=E∩f−1(A) and H=E∩f−1(B). Then Rudin is claiming that E=G∪H. I'm a little suspicious about this. What if f is non-surjective, then f−1(A)∪f−1(B) is only a proper subset of E? Is there a property of f being continuous that forces f to be 1-1?