Let us learn MAKS

  Thomae's function is a real-valued function of a real variable that can be defined as:       is continuous at all irrational numbers. But, is discontinuous at all rational numbers.  It is interesting to note that there doesn't exist any function f : R → R that is continuous only at the rational points. WHY?  Suppose such a function exists, then R can be written as a countable union of nowhere dense sets, contradicting the Baire category theorem. f ( x ) = { 1 q if  x = ...