Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / E / 2 / ii

If $f(A)$ had a separation into two nonempty relatively open disjoint pieces, their inverse images under the restriction $f{|}_A$ would be two nonempty relatively open disjoint subsets covering $A$. This would contradict connectedness of $A$. Thus a continuous image of a connected set is connected.