## Functional equation

For discussing Olympiad Level Algebra (and Inequality) problems
anando
Posts: 3
Joined: Mon Sep 07, 2020 3:37 pm

### Functional equation

$\ f: \mathbb{R} \rightarrow \mathbb{R}$
$\ f(x+y)=f(x)+f(y))$

Atonu Roy Chowdhury
Posts: 64
Joined: Fri Aug 05, 2016 7:57 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).