Characterizing continuity by preserving compactness and connectedness

Let us call a function f from a space X into a space Y preserving if the image of every compact subspace of X is compact in Y and the image of every connected subspace of X is connected in Y. By elementary theorems a continuous function is always preserving. McMillan [Pacific J. Math. 32 (1970) 479] proved in 1970 that if X is Hausdorff, locally connected and Frechét, Y is Hausdorff, then the converse is also true: any preserving function f :X→Y is continuous. The main result of this paper is that if X is any product of connected linearly ordered spaces (e.g., if X=Rκ) and f :X→Y is a preserving function into a regular space Y, then f is continuous.