  We propose a notion of continuous path for locally finite metric spaces,
taking inspiration from the recent development of A-theory for locally finite
connected graphs. We use this notion of continuity to derive an analogue in Z^2
of the Jordan curve theorem and to extend to a quite large class of locally
finite metric spaces (containing all finite metric spaces) an inequality for
the \ell^p-distortion of a metric space that has been recently proved by
Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.
