Please solve this functional equation:

$\ f: \mathbb{R} \rightarrow \mathbb{R}$

$\ f(x+y)=f(x)+f(y))$

## Functional equation

- Atonu Roy Chowdhury
**Posts:**64**Joined:**Fri Aug 05, 2016 7:57 pm**Location:**Chittagong, Bangladesh

### Re: Functional equation

The equation $f(x+y)=f(x)+f(y)$ is generally called

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:

**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).

This was freedom. Losing all hope was freedom.