Spivak Thomae's Function proof explanation

Thread starter Alpharup
Start date Dec 19, 2015
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Explanation Function Proof Spivak

In summary, @mathwonk has shown that the limit of f(x) as x tends to a(it is between 0 and 1) is 0. He has also shown that there is a ¿(delta) for which 0<|x-a|<¿.

Jan 15, 2016

Samy_A

Science Advisor

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Alpharup said:

I can't understand this step

##A<\epsilon## means that ##A## is smaller than ##\epsilon##.
##A\leq \epsilon## means that ##A## is smaller than or equal to ##\epsilon##.
Any ##A## that satisfies ##A< \epsilon## will also satisfy ##A\leq \epsilon##.

That's all that there is to that step.

<h2>1. What is Spivak Thomae's Function?</h2><p>Spivak Thomae's Function, also known as the Thomae Function or the Popcorn Function, is a mathematical function that is defined as f(x) = 0 for irrational numbers and f(x) = 1/n for rational numbers, where n is the smallest integer such that x can be written as a fraction with a denominator of n.</p><h2>2. What is the significance of Spivak Thomae's Function?</h2><p>Spivak Thomae's Function is significant because it is a counterexample to many common conjectures in mathematics, such as the Intermediate Value Theorem and the Continuity Theorem. It also serves as an example of a function that is continuous at irrational numbers but discontinuous at rational numbers.</p><h2>3. What is the proof for Spivak Thomae's Function?</h2><p>The proof for Spivak Thomae's Function involves showing that for any given rational number, there exists a neighborhood around it where the function takes on values close to 1, and for any given irrational number, there exists a neighborhood around it where the function takes on values close to 0. This can be done using the Archimedean Property and the Density of Rational Numbers.</p><h2>4. How does Spivak Thomae's Function relate to the concept of continuity?</h2><p>Spivak Thomae's Function is an example of a function that is continuous at irrational numbers but discontinuous at rational numbers. This means that it satisfies the definition of continuity at all irrational numbers, but not at any rational numbers. This highlights the importance of specifying the domain of a function when discussing continuity.</p><h2>5. Are there any real-world applications of Spivak Thomae's Function?</h2><p>While Spivak Thomae's Function may not have direct real-world applications, it serves as a valuable tool for understanding the concepts of continuity and differentiability in mathematics. It also highlights the importance of carefully defining functions and considering their domains when making mathematical statements.</p>

1. What is Spivak Thomae's Function?

Spivak Thomae's Function, also known as the Thomae Function or the Popcorn Function, is a mathematical function that is defined as f(x) = 0 for irrational numbers and f(x) = 1/n for rational numbers, where n is the smallest integer such that x can be written as a fraction with a denominator of n.

2. What is the significance of Spivak Thomae's Function?

Spivak Thomae's Function is significant because it is a counterexample to many common conjectures in mathematics, such as the Intermediate Value Theorem and the Continuity Theorem. It also serves as an example of a function that is continuous at irrational numbers but discontinuous at rational numbers.

3. What is the proof for Spivak Thomae's Function?

The proof for Spivak Thomae's Function involves showing that for any given rational number, there exists a neighborhood around it where the function takes on values close to 1, and for any given irrational number, there exists a neighborhood around it where the function takes on values close to 0. This can be done using the Archimedean Property and the Density of Rational Numbers.

4. How does Spivak Thomae's Function relate to the concept of continuity?

Spivak Thomae's Function is an example of a function that is continuous at irrational numbers but discontinuous at rational numbers. This means that it satisfies the definition of continuity at all irrational numbers, but not at any rational numbers. This highlights the importance of specifying the domain of a function when discussing continuity.

5. Are there any real-world applications of Spivak Thomae's Function?

While Spivak Thomae's Function may not have direct real-world applications, it serves as a valuable tool for understanding the concepts of continuity and differentiability in mathematics. It also highlights the importance of carefully defining functions and considering their domains when making mathematical statements.