Fixed-point free maps on the reals and more

In our main result we prove that a continuous function f : R → R fixes a point iff f ˜ : β R → β R does, where f ˜ is the continuous extension of f over the Čech–Stone compactification. In the second half of the paper the statement of this theorem with R replaced by an arbitrary X is taken for a definition of a property and a number of results about the property are proved.