## IMC 2016 – Day 1 – Problem 3

**Problem 3.** Let be a positive integer. Also let and be reap numbers such that for . Prove that

## 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.

## 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 Problem with two solutions

Suppose is continuous and . Prove that if the Cauchy Problem has two distinct solutions then it has infinitely many solutions.

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## 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 .