Page 1 of 1
Functional equation
Posted: Sat Oct 24, 2020 5:11 pm
by anando
Please solve this functional equation:
$\ f: \mathbb{R} \rightarrow \mathbb{R}$
$\ f(x+y)=f(x)+f(y))$
Re: Functional equation
Posted: Thu Dec 03, 2020 11:35 pm
by Atonu Roy Chowdhury
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:
https://web.evanchen.cc/handouts/FuncE ... -Intro.pdf