Table of Contents

## Is Dirichlet function Riemann integrable?

The Dirichlet function is discontinuous everywhere, and you can use it to construct functions that are only continuous at a single point. It is also a standard example of a function that is not Riemann integrable (although it is Lebesgue integrable).

**Is the modified Dirichlet function integrable?**

On the other hand, the upper integral of Dirichlet function is b−a, while the lower integral is 0. They don’t match, so that the function is not Riemann integrable.

**How do you prove a Dirichlet is discontinuous?**

Let D:R→R denote the Dirichlet function: ∀x∈R:D(x)={c:x∈Qd:x∉Q. where Q denotes the set of rational numbers. Then D is discontinuous at every x∈R.

### What is Dirichlet formula?

In many situations, the dissipation formula which assures that the Dirichlet integral of a function u is expressed as the sum of -u(x)[Δu(x)] seems to play an essential role, where Δu(x) denotes the (discrete) Laplacian of u. This formula can be regarded as a special case of the discrete analogue of Green’s Formula.

**Why is the Dirichlet function nowhere continuous?**

Because this oscillation cannot be decreased by making the neighborhood smaller, there is no limit at a, not even one-sided. Since we do not have limits, we also cannot have continuity (even one-sided), that is, the Dirichlet function is not continuous at a single point.

**Can the Dirichlet function be a derivative?**

Since we do not have limits, we also cannot have continuity (even one-sided), that is, the Dirichlet function is not continuous at a single point. Consequently we do not have derivatives, not even one-sided. There is also no point where this function would be monotone.

#### Why is the Dirichlet function discontinuous?

**What is Dirichlet boundary value problem?**

In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.

**What is Cauchy problem in PDE?**

The Cauchy problem consists of finding the unknown function(s) u that satisfy simultaneously the PDE and the conditions (1.29). The conditions (1.29) are called the initial conditions and the given functions f0,f1,…,fk−1, will be referred to as the initial data.

## Why is Dirichlet function discontinuous?

Topological properties The Dirichlet function is nowhere continuous. If y is rational, then f(y) = 1. To show the function is not continuous at y, we need to find an ε such that no matter how small we choose δ, there will be points z within δ of y such that f(z) is not within ε of f(y) = 1. In fact, 1/2 is such an ε.

**Is Dirichlet function a simple function?**

A basic example of a simple function is the floor function over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise.