Is the function f: R -> R, x -> x^2 continuous when the domain and codomain are given the Half interval topology? (Or Lower Limit topology). I'm not sure where to go with this. On inspection, I know that the intervals are open sets, so preservance of open sets in preimages are defined for x > 0. But what if there is a set [x^2,x^2+r) that is in the negative part of the real line, there is no pre-image for this set. Is there something I'm missing, or just not realizing (most likely the second one)?