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

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 $f(A)$ is connected.