Question

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

Answer 1

If e∈E, then f(e)∈f(E)=A∪B.

So, f(e)∈A or f(e)∈B.

So, e∈f−1(A) or e∈f−1(B).

So, e∈f−1(A)∪f−1(B).

So, E⊂f−1(A)∪f−1(B).

So, E=E∩(f−1(A)∪f−1(B))=(E∩f−1(A))∪(E∩f−1(B))=G∪H.

A∩B=∅.

So, f−1(A)∩f−1(B)=∅.

So, ∅=E∩∅=E∩(f−1(A)∩f−1(B))=(E∩f−1(A))∩(E∩f−1(B))=G∩H