## Sierpinski’s Theorem for Additive Functions – Simplification

1. If is a solution of the Cauchy functional equation which is surjective, but not injective, then has the Darboux property.

2. For every solution of the Cauchy functional equation there exist two non-trivial solutions of the same equation, such that and have the Darboux property and .

These two results were proven in this post. The version presented here is a simplified one, identifying exactly what we need in order to obtain the desired results.

## Sierpinski’s Theorem for Additive Functions

We say that is an additive function if

1. Prove that there exist additive functions which are discontinuous with or without the *Darboux Property*.

2. Prove that for every additive function there exist two functions which are additive, have the *Darboux Property*, and .

The second part is similar to Sierpinski’s Theorem which states that every real function can be written as the sum of two real functions with Darboux property.

(A function has the *Darboux property* if for every , is an interval.)

## Miklos Schweitzer 2013 Problem 7

**Problem 7.** Suppose that is an additive function (that is for all ) for which is bounded of some nonempty subinterval of . Prove that is continuous.

## Cauchy Functional Equation

We say that a function satisfies the **Cauchy** functional equation if .

1. Prove that . ( There fore, is a linear application, of we consider considered a vector space over .)

2. The following statements are equivalent:

i) .

ii) is continuous.

iii) there exists a point such that is continuous in .

iv) is non-decreasing/non-increasing.

v) is bounded on some interval.

vi) is positive/negative when is positive/negative.

**Solutions of the type above are called “trivial” solutions of the Cauchy functional equation.**

3. Prove that there exist non-trivial solutions of the Cauchy functional equation.

4. Prove that if is a non-trivial solution of the Cauchy functional equation then , for any .

5. Prove that there exist solutions of the Cauchy functional equation which have the Darboux property, which means that is an interval for any interval .