You will be surprised to know that the concept of function is not very old. Till 1837 it was believed that every function could be traced with "free motion of the hand". A mathematician Gustav Dirichlet in 1837 gave the definition of function which we use most often now. He also gave the example the function is irrational and is a rational. (This is the indicator function of the set of rationals in). Do you think you can graph the function?
is irrational and 
is a rational. (This is the indicator function of the set of rationals in
). Do you think you can graph the function? 
