Are there real numbers that are neither rational nor irrational? I wouldn't have asked this question if I hadn't seen this image:

From this image it seems like there are reals that are neither rational nor irrational (dark blue), but is it so or is that illustration incorrect?

We can represents real numbers on line i.e. real line which contains rationals and irrationals. Now by completeness property of real numbers, which says that real line has no gap. So there is no real number that is neither rational nor irrational. 
