## 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.)

## Darboux Functions with no Iterate Fixed Points

It is well known that there exist functions which have Darboux property and they have no fixed points. An example can be found in an earlier post of mine. Here is a generalization of that result.

There exist functions which have Darboux property and for which none of its iterates has a fixed point, i.e. for every and for every .

## Contest problem

There exist functions with Darboux property, such that there exist sets with and .

*Problem proposed in the “Traian Lalescu” contest for undergraduate students in Romania in 2004.*

## Sierpinski’s Theorem

For any function there exist two functions such that and have the **Darboux property**.

A function has the **Darboux property** if for any interval we have is also an interval. This is slightly different from continuity and intermediate value property. Cotinuity implies Darboux and Darboux implies Intermediate value property.

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