Technically, both are not discontinuous. 1 is the simplified form of 1.000..., and there is no difference at all between an infinite number of zeroes and an infinite number of 9s (in terms of the way they are treated).
Both 1.000... and 0.999... are real (and rational) numbers, and they must abide by the rules of rational / real numbers. That includes being complete, which means that 0.999... is not simply a limit, but at actual value, equal to the value of 1.000...
Both 1.000... and 0.999... are real (and rational) numbers, and they must abide by the rules of rational / real numbers. That includes being complete, which means that 0.999... is not simply a limit, but at actual value, equal to the value of 1.000...