Let f be a one-to-one function from a compact Hausdorff space X onto a compact Hausdorff space Y. Show that if f[K] is compact for every compact K subset of X, then f is continuous.
I know that f restricted to K must be continuous, because the continuous image of a compact set is compact. I'm not sure how to prove that f is continuous, other than the fact that every b in Y has a distinct f(b) in X because f is one-to-one and onto.
I'm not sure how to set this problem up.