For a fixed real number x, the map defined by is a continuous bijection. It follows that if S is dense and open then so is . In words, if S is dense and open then so is the reflected translate of S by x. The same is therefore true for the complement of S. So reflected translates of nowhere dense sets are nowhere dense. The same is therefore also true for countable unions of such sets, so reflected translates of first category sets are of first category. Taking complements again, you see that reflected translates of residual sets are residual.

Similarly, the map defined by is a continuous bijection. So you can use the same argument as for translates to show that if is residual then so is .