at the edge of town - am Stadtrand
Rand rather means border, margin, rim.
edge of a knife - Schneide eines Messers,
to teeter on a knife edge - auf Messers Schneide stehen
edge of a cliff - am Rande des Abgrunds
Kante is a mathematical and technical term. (edge to edge - Kante an Kante) You use it as well for Tischkante (edge of a table).
living on the edge - gefaehrlich leben
Heh, it's the hook on an old Aerosmith song "Living on the Edge" ...just a dumb thing that if I left the word for edge out of, I'd still know what to fill in the blank for: "Wir leben an _____!"
In checking out the song in more detail, it does talk about falling off that edge, so maybe "Rand" is the best choice after all!
Thanks!