I am defining continuity such that single point that is undefined does not break continuity if and only if it has the same limit when approached from both sides. This is because, as I say, a point is arbitrarily small and therefore if the limit is the same on both sides, it's simpler to treat it as still a continuous function.

Otherwise I think you get some problems which break algebra, because multiplying of dividing by variables can break the domains of functions.

If we have f(x) = 1, then this should be the same even if we multiply both sides by x. x * f(x) = x, which results in f(x) = x/x. You break algebra by allowing such straight-forward manipulations (which really are the core of algebra) to tamper with the original domain of the function.

Just as you can't just treat a square root as a core algebraic process (because it is not a function), you would no longer be able to divide by a variable, since that would be undefined wherever the variable is 0.

Edit Also it occurs to me that treating x/x as undefined at 0 would also man that the first derivative of f(x) = x would also be undefined at 0, which can't work either.

So therefore treating x/x as discontinuous at 0 breaks both algebra and calculus.