And why do you mention only rational numbers? In order for a function to be continous at a point, its limit must exist and be equal to the value of the function. But, given any x, there will be rational numbers as close as we please to x and so values close to x itself. There will also be IRRATIONAL numbers arbitrarily close to x so there will be values close to 1- x. In order that the limit exist, we must have x= 1- x. That's what is special about 1/2! I'm sure that's what you were thinking but you need to state clearly why this would work with 1/2 only and include the irrational numbers.