# Nowhere Continuous Function Dirichlet Proof

• cmajor47
In summary, the Nowhere Continuous Function Dirichlet Proof shows that there exists a real-valued function that is continuous only at irrational points and discontinuous at all rational points. This proof is an important result in real analysis and can be used to further understand the properties of continuous functions. It also demonstrates the existence of functions that do not follow the intuitive definition of continuity and highlights the importance of rigorous mathematical proofs in understanding mathematical concepts.
cmajor47

## Homework Statement

Prove that the Dirichlet function is continuous nowhere.

## Homework Equations

Dirichlet function = 1 when x is rational, and 0 when x is irrational.

## The Attempt at a Solution

I was looking at this proof on http://math.feld.cvut.cz/mt/txtd/1/txe4da1c.htm
At the very end when the creator shows that inf f(x) $$\neq$$ sup f(x)
how does this tell you that the function is never continuous?
Do the greatest lower bound and least upper bound of a function have to be equal at some point for a function to be continuous?

cmajor47 said:
At the very end when the creator shows that inf f(x) $$\neq$$ sup f(x)
That's not what he showed.

He showed the infimum (over all P) of U(f,P) was not the supremum (over all P) of L(f,P).

And he wasn't proving anything about continuity anyways; that article is about Riemann integrability.

Oh, wow, don't know how I missed that. Thanks!

## 1. What is a nowhere continuous function?

A nowhere continuous function is a mathematical function that is not continuous at any point in its domain. This means that there is no single point at which the function can be defined as continuous.

## 2. What does it mean for a function to be Dirichlet proof?

A function is Dirichlet proof if it satisfies the Dirichlet conditions, which state that the function is bounded, has a finite number of discontinuities, and has a finite number of maxima and minima within any given interval.

## 3. How is the proof for a nowhere continuous function using Dirichlet's theorem?

The proof for a nowhere continuous function using Dirichlet's theorem involves showing that the function satisfies the Dirichlet conditions, and therefore can be considered Dirichlet proof. This helps to establish that the function is not continuous at any point in its domain.

## 4. Can a nowhere continuous function have a limit at any point?

No, a nowhere continuous function cannot have a limit at any point. This is because limits require the function to be continuous at that point, which a nowhere continuous function is not.

## 5. What are some examples of nowhere continuous functions?

Some examples of nowhere continuous functions include the Dirichlet function, the Cantor function, and the Weierstrass function. These functions are commonly used in mathematical proofs and are known for their lack of continuity at any point in their domain.

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