If $f(A)$ were separated by two disjoint nonempty relatively open sets, their preimages under the restriction $f|_A$ would be disjoint nonempty relatively open sets covering $A$. <Continuity> makes them open in $A$, contradicting its <connectedness>. Therefore \b[$f(A)$ is <connected>].

Solution

Past exam of the options I course of the University of Oxford / 2008 / E / E2 / ii / Solution

 Solution (source code)