User blog comment:B1mb0w/FGH of Omega/@comment-24920136-20160607065741/@comment-1605058-20160607071816

If we define $$f_0$$ to be successor, then $$f_0(\omega)=\omega+1$$ and there is no reason to expect this function to preserve limits. The successor function $$\alpha\mapsto\alpha+1$$ is not continuous.