Editing Nonstandard calculus (section)

==Example: Dirichlet function==
Consider the [[Dirichlet function]]
:<math>I_Q(x):=\begin{cases} 1 & \text{ if }x \text{ is rational}, \\
0 & \text{ if } x \text{ is irrational}. \end{cases}</math>

It is well known that, under the [[continuous function|standard definition of continuity]], the function is discontinuous at every point.  Let us check this in terms of the hyperreal definition of continuity above, for instance let us show that the Dirichlet function is not continuous at π.  Consider the continued fraction approximation a<sub>n</sub> of π.  Now let the index n be an infinite [[hypernatural]] number.  By the [[transfer principle]], the natural extension of the Dirichlet function takes the value 1 at a<sub>n</sub>.  Note that the hyperrational point a<sub>n</sub> is infinitely close to π.  Thus the natural extension of the Dirichlet function takes different values (0 and 1) at these two infinitely close points, and therefore the Dirichlet function is not continuous at&nbsp;''π''.