Dirichlet's Function: Why is it Difficult to Draw the Graph?

[Pyroadept]

Homework Statement

Define Dirichlet's function f by putting f(x) = 1 if x is rational and f(x) = 0 if x is irrational. Explain why it is difficult to draw the graph of f. Prove that the lower Riemann sum L(x_0,...,x_n) is always equal to 0 and the upper Riemann sum U(x_0,...x_n) is always equal to 1.

Homework Equations

Equations for upper and lower Riemann sums.

The Attempt at a Solution

Hi everyone,
Here's what I've done so far:

The graph is difficult to draw because there are infinitely many rational numbers and infinitely many irrational numbers, all interspersed among one another, so you will continuously be switching between f(x) = 0 and f(x) = 1.

For every rational number, there is an irrational number, so any chosen interval will contain both a rational [f(x) = 1] and irrational [f(x) = 0] number.
m = 0 and M = 1
So the lower Riemann sum will be zero, as two points side-by-side (i.e. a rational and an irrational with only 'vertical' area between them) will have minimum vertical area 0.
And, for the upper Riemann sum, two points side-by-side will have maximum vertical area 1.



Is this correct?

Thanks for any help

 

[benorin]

Yep, graph difficulty is correct. The graph would look like two solid lines, y=1 and y=0.

The second part is close. Between every two rational numbers lies an irrational number. In fact, the irrationals are dense in the reals. Furthermore, between every two irrationals is a rational, and generally between any two real numbers there are both irrational and rational numbers. And you got the rest.