Example of Continuity of Real Valued Function Dependent on Topology

The real valued functionis given by

is not continuous if setr is given the ordinary Euclidean topology with metricconsisting of open intervals of the formbut is continuous ifhas the upper limit topology . The upper limit topology consists of unions of half open sets of the form

Letbe the open setinso the inverse image of an open set is not an open set andis not continuous with the Euclidean topology.

The upper limit topology onconsists of all half open intervals of the form

The inverse image of an open set is an open set nis an open set sof is continuous with this topology.