Functional equation

For discussing Olympiad Level Algebra (and Inequality) problems
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Functional equation

Unread post by anando » Sat Oct 24, 2020 5:11 pm

Please solve this functional equation:
$\ f: \mathbb{R} \rightarrow \mathbb{R}$
$\ f(x+y)=f(x)+f(y))$

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Atonu Roy Chowdhury
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Re: Functional equation

Unread post by Atonu Roy Chowdhury » Thu Dec 03, 2020 11:35 pm

The equation $f(x+y)=f(x)+f(y)$ is generally called Cauchy Functional Equation. Sadly, we can't find an explicit solution for Cauchy Functional Equation when the domain is Real (if no other conditions are given). However we can find solutions when the domain is Natural numbers or Rational numbers. The solution is \[f(x)=kx \text{ where } k \text{ is a constant.}\]
But hey, don't be disappointed. Cauchy Functional Equations can be solved over $\mathbb{R}$ if at least one of the following conditions are satisfied:
  • $f$ is continuous in any interval.
  • $f$ is bounded (either above or below) in any nontrivial interval.
  • There exists $(a, b)$ and $\varepsilon >0$ such that $(x - a)^2 + (f(x) - b)^2 > \varepsilon$ for every $x$. (i.e. the graph of $f$ omits some disk, however small).
You can read about introductory functional equations form this wonderful note written by Evan Chen: ... -Intro.pdf
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