Making the domain of f(x)=1/x be the non-zero real numbers allows f to be continuous. If f(0)=0 then continuity is lost. Alternatively, defining f(0) to be inf

Making the domain of f(x)=1/x be the non-zero real numbers allows f to be continuous. If f(0)=0 then continuity is lost. Alternatively, defining f(0) to be infinity (a projectively extended real number) also allows f to be continuous. Another possibility is to define f(0) to be some kind of "bottom" value. See the following tweets by Andrej Bauer: https://twitter.com/andrejbauer/status/1268860981943439361 https://twitter.com/andrejbauer/status/1268860981943439361 Continuity is important because without it we lose computability. Every computable function is continuous (in an appropriate sense).