Is hyperbola xy=1, a continuous function?

Descartes said that a function is continuous if its graph can be drawn without lifting the pencil from the paper. If xy=1, it means y=1/x. Then is '1' the point of symmetry here? xy=1 tells us that since we cannot reach infinity we can never reach 0. Is duration between 1 and 0 equal that between 1 and infinity in a hyperbola? Does our number system have two points of symmetry: one at 0 and the other at 1? Can we show xy=1 using Cartesean coordinates?