A function discontinuous at every point. Use the fact that between any two distinct real numbers there exist (infinitely many) rational numbers and (infinitely many) irrational numbers to show that the function (f(x) = 0, if x is rational and 1, if x is irrational) is discontinuous at every point. To gain intuition about this function, think about how you would draw it's graph. The only reasonable way that I know to draw the graph of this function is as two horizontal lines, one at height 0 and one at height 1 (but the function still passes the vertical line test because the horizontal lines have invisible holes in them in just the right places!). The function values are jumping up and down between 0 and 1 so fast that the function looks like it is taking both values at the same time; much in the same way that the rotor blades on a helicoptor appear to be in more than one place at the same time when they are spinning rapidly.

In calculus, the notion of "area under the graph" of a function y = f(x) is fundamental. How would one define/compute the area under the graph of Dirichlet's function, say between x = 0 and x = 1? Should this area be 0, 1, 1/2 , or none of these? (This problem confounded mathematicians for some time. Dirichlet's function may or may not be relevant to real life, but it lead to some important developments in advanced calculus that are.)