Non-standard Calculus - Example: Dirichlet Function

Example: Dirichlet Function

Consider the Dirichlet function

I_Q(x)=\begin{cases} 1 & \text{ if }x \text{ is rational}, \\ 0 & \text{ if } x \text{ is irrational}. \end{cases}

It is well known that 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 an 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 an. Note that the hyperrational point an 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 π.

Read more about this topic:  Non-standard Calculus

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