Past exam of the options II course of the University of Oxford / 2007 / E / E3 / d

False. Take $X=[-1,1]$, $Y=[0,1]$ and $f(x)=x^2$, with their usual topologies. The domain is compact and connected, the codomain is Hausdorff, and the map is continuous and onto. But the fibre over one is $\{-1,1\}$, a disconnected two-point subspace.