Well this gets even better yet. For a function such as y =1/x we need the first macroinfinity number which is 1*10^603 + 1*10^-603. So a tiny bit beyond the last finite number of 1*10^603 starts the first infinity number and it patches up any functions such as y =1/x when x =0. But for a function like that of y = sin(x), we do not have to go out to 1*10^603 + 1*10^-603, but rather and infinity number in between 0 and 1*10^-603 or in between any two successive numbers in the empty space region of those successive numbers.
So in New Math, there is no such thing as a discontinuous function, because the empty space between successive finite numbers lies infinity numbers and these infinity numbers eliminate division by 0 as undefined and makes operation by an infinity number undefined.
So you have a function graphed in Old Math and you have what appears to be a discontinuity, then all you do is take a infinity number between those two successive finite numbers and enlist it or volunteer it as the connector to the graph.