Posts Tagged ‘darboux’

Sierpinski’s Theorem for Additive Functions

January 26, 2014 1 comment

We say that {f: \Bbb{R} \rightarrow \Bbb{R}} is an additive function if

\displaystyle f(x+y)=f(x)+f(y),\ \forall x,y \in \Bbb{R}.

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

2. Prove that for every additive function {f} there exist two functions {f_1,f_2:\Bbb{R} \rightarrow \Bbb{R}} which are additive, have the Darboux Property, and {f=f_1+f_2}.

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 {g:I \rightarrow \Bbb{R}} has the Darboux property if for every {[a,b]\subset I}, {g([a,b])} is an interval.)

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Darboux Functions with no Iterate Fixed Points

June 3, 2012 Leave a comment

It is well known that there exist functions {f:[0,1] \rightarrow [0,1]} 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 {f:[0,1] \rightarrow [0,1]} which have Darboux property and for which none of its iterates has a fixed point, i.e. {\underbrace{f\circ ..\circ f}_{n \text{ times}}(x)\neq x} for every {x \in [0,1]} and for every {n \geq 1}.

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Contest problem

September 8, 2009 2 comments

There exist functions f:[0,1]\to [0,1] with Darboux property, such that there exist sets A,B with A\cap B=\emptyset,\ A\cup B=[0,1] and f(A)\subseteq B,\ f(B)\subseteq A.
Problem proposed in the “Traian Lalescu” contest for undergraduate students in Romania in 2004.

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Categories: Analysis, Olympiad, Undergraduate Tags:

Sierpinski’s Theorem

September 8, 2009 5 comments

For any function f:\mathbb{R} \to \mathbb{R} there exist two functions f_1, \ f_2 such that f=f_1+f_2 and f_1,f_2 have the Darboux property.
A function has the Darboux property if for any interval I\subseteq \mathbb{R} we have f(I) 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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