## Does the Dirichlet function converge?

1.8 Corollary : For η(s) or for L(s, χ) with χ any non-trivial Dirichlet character χ modulo q, the Dirichlet series converges for σ > 0. We remark that Theorem 1.7 is proved in [4], but the proof is a little less direct than the one given above, and is restricted to the case an ∈ R (for no apparent reason).

**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 kind of discontinuities does Dirichlet function have?

The classification you link to defines an “infinite discontinuity” to mean the case where “one or both of the limits L− and L+ does not exist or is infinite”. That’s precisely what happens with the Dirichlet function everywhere.

**Is Thomae’s function continuous?**

Thomae Function is Continuous at Irrational Numbers.

## 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.

**How do you solve a Dirichlet problem?**

Methods of solution For bounded domains, the Dirichlet problem can be solved using the Perron method, which relies on the maximum principle for subharmonic functions.

### Are Dirichlet functions integrable?

The Dirichlet function is Lebesgue-integrable on R and its integral over R is zero because it is zero except on the set of rational numbers which is negligible (for the Lebesgue measure).

**Is the Dirichlet function Riemann 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.

## Why is the Dirichlet function not continuous?

The Dirichlet function is nowhere continuous, since the irrational numbers and the rational numbers are both dense in every interval [ a, b]. On every interval the supremum of f is 1 and the infimum is 0 therefore it is not Riemann integrable.

**What is the Dirichlet function in math?**

In mathematics, the Dirichlet function is the indicator function 1 ℚ of the set of rational numbers ℚ, i.e. 1 ℚ (x) = 1 if x is a rational number and 1 ℚ (x) = 0 if x is not a rational number (i.e. an irrational number). It is named after the mathematician Peter Gustav Lejeune Dirichlet.

### Is the Dirichlet function a Baire class 2 function?

The Dirichlet function is an archetypal example of the Blumberg theorem. for integer j and k. This shows that the Dirichlet function is a Baire class 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a meagre set.