  Among the coordinates used to construct a conformal compactification of the
Schwarzschild spacetime, none of them simultaneously extend smoothly both
through an event horizon and beyond null infinity.To construct such
coordinates, instead of starting with the Kruskal-Szekeres coordinates we
assume direct analytic transformation between Schwarzschild and compactified
coordinates and determine their behavior on the event horizon and at null
infinity. We then propose an example of such coordinates and illustrate the way
they cover the conformally extended Schwarzschild spacetime as well as their
suitability for numerical applications.
