Your reasoning isn't very clear. Yes, given a not equal to 1/2 we choose some rational number between x and a- but that is true with any number replacing 1/2! What's special about 1/2?

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 willalsobe 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'swhat 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.