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

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In summary, Dirichlet's function is difficult to graph due to the infinite amount of rational and irrational numbers it includes. The lower Riemann sum is always equal to 0 and the upper Riemann sum is always equal to 1 because between any two points on the graph, there will always be both a rational and irrational number, resulting in a minimum and maximum vertical area of 0 and 1, respectively.
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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
 
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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.
 

1. What is the Dirichlet Function?

The Dirichlet function, denoted by D(x), is a mathematical function that is defined as follows: D(x) = 1 if x is rational, and D(x) = 0 if x is irrational. Essentially, the function assigns a value of 1 to rational numbers and a value of 0 to irrational numbers.

2. When was the Dirichlet Function discovered?

The Dirichlet function was first introduced by German mathematician Peter Gustav Lejeune Dirichlet in the 19th century. However, it was not until the 20th century that it gained widespread recognition and became a fundamental concept in mathematical analysis.

3. What are the properties of the Dirichlet Function?

The Dirichlet function has several interesting properties. It is discontinuous at every point, meaning that it cannot be represented by a single continuous curve. Additionally, it is non-integrable, which means that it cannot be integrated over any interval.

4. How is the Dirichlet Function used in mathematics?

The Dirichlet function is primarily used in mathematical analysis and topology. It is a simple yet powerful tool for proving the existence of certain types of functions, and it has applications in various areas of mathematics such as measure theory and the study of fractals.

5. Are there any real-world applications of the Dirichlet Function?

While the Dirichlet function does not have direct real-world applications, it has been used in the field of computer graphics to generate images with irregular patterns. It has also been used in cryptography to create secure hash functions. Overall, the function's significance lies in its theoretical and foundational role in mathematics.

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