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Sierpinski’s Theorem for Additive Functions – Simplification

February 3, 2014 Leave a comment

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

2. For every solution {f} of the Cauchy functional equation there exist two non-trivial solutions {f_1,f_2} of the same equation, such that {f_1} and {f_2} have the Darboux property and {f=f_1+f_2}.

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.

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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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Miklos Schweitzer 2013 Problem 7

November 22, 2013 1 comment

Problem 7. Suppose that {f: \Bbb{R} \rightarrow \Bbb{R}} is an additive function (that is {f(x+y) = f(x)+f(y)} for all {x, y \in \Bbb{R}}) for which {x \mapsto f(x)f(\sqrt{1-x^2})} is bounded of some nonempty subinterval of {(0,1)}. Prove that {f} is continuous.

Miklos Schweitzer 2013

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Cauchy Functional Equation

October 1, 2009 Leave a comment

We say that a function f: \mathbb{R} \to \mathbb{R} satisfies the Cauchy functional equation if f(x+y)=f(x)+f(y),\ \forall x,y \in \mathbb{R}.
1. Prove that f(qx)=qf(x),\ \forall q \in \mathbb{Q},\ \forall x \in \mathbb{R}. ( There fore, f is a linear application, of we consider \mathbb{R} considered a vector space over \mathbb{Q}.)
2. The following statements are equivalent:
i) \exists a \in \mathbb{R},\ such \ that \ f(x)=ax,\ \forall x \in \mathbb{R}.
ii) f is continuous.
iii) there exists a point x_0 such that f is continuous in x_0.
iv) f is non-decreasing/non-increasing.
v) f is bounded on some interval.
vi) f(x) is positive/negative when x 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 f is a non-trivial solution of the Cauchy functional equation then \overline{f([a,b])}=\mathbb{R}, for any a,b \in \mathbb{R},\ a<b.
5. Prove that there exist solutions of the Cauchy functional equation which have the Darboux property, which means that f(I) is an interval for any interval I\subset \mathbb{R}.

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