This function is impossible to graph. The picture above is only a poor representation of the true graph. Nonetheless, take an arbitrary point on the real axis. We can find a sequence { } of rational points that converge to from the right. Then g( ) converges to 1. But we can also find a sequence { } of irrational points converging to from the right. In that case g( ) converges to 0. But that means that the limit of g(x) as x approaches from the right does not exist. The same argument, of course, works to show that the limit of g(x) as x approaches from the left does not exist. Hence, is an essential discontinuity for g(x).