Defintion:                 1, if x is a rational number
                    f(x)= {
                                 0, if x is an irrational number

~Let's discuss rational & irrational numbers a bit.

~a rational number is any number that can be expressed as the quotient (ratio) of two integers (includes negatives).

~If the number can't, then it is irrational.

~zero is rational since it can be expressed as 0 over any non-zero integer.

~p, e, & roots of numbers that are not exact (i.e., √(2)) are examples of irrational numbers. They are unending, non-repeating decimals. Since  there are no predictable patterns to the digits in decimal form, these numbers serve as excellent data bases for random digits.

~between any two rational numbers is another rational number.

~between any two irrational numbers is another irrational number.

~between any two rational numbers is an irrational number.

~between any two irrational numbers is a rational number.

~However, the infinity sizes of rationals & irrationals are different.
 (a topic for a later discussion)

~Our "Holey function" has a domain of all real numbers. Therefore,
there are infinitely many rational values and irrational values for x.

~For each rational value for x, the functional value is 1.
  For each irrational value for x, the functional value is 0.

~Either, a vertical line cuts the curve at level 1 or at level 0. Hence, it's a perfectly defined function.

~If x is rational, then there is a hole at  level 0 and a point at level 1.

~If x is irrational, then there is a hole at level 1 and a point at level 0.

~Hence, the motivation for its name.

~This function is discontinuous at every x value. As we approach any fixed x on the x axis, no limit exists. The function oscillates between 0 and 1 no matter how close we get to that value.

~Therefore, it does not have a derivative anywhere (totally non-differentiable).

~If you attempted to sketch its graph, it would appear as two horizontal lines, level 1 and level 0, however, these lines would be full of holes which you could not see. So, it does satisfy the verical line test for a function of x.

~There is no area defined under its graph (see GLB & LUB link) since the LUB of all lower sums does not equat the GLB of all upper sums.

~So, what good is it?  

~By studing such a curve that misbehaves so much, we gain insight and, consequently, a better understanding of the concepts discussed.
Sometimes, to fully understand a concept, one should examine situations where it doesn't hold. (i.e., to understand continuity, one should study & understand cases where functions are discontinuous)