Provide all continuous ${ }^{1}$ solutions of the Cauchy equation on $\mathbb{R}$.
By passing from broad inequalities to the limit, if $f$ is continuous and if $f_{\mathbb{Q}}$ is monotone, then $f$ is monotone. Since $f_{|\mathbb{Q}|}$ is of the form $x \mapsto a x$ according to the preceding, $f$ is linear.

Conclusion The overall strategy we have implemented here is to solve the equation over increasingly larger domains, taking advantage of the principle that if $f$ is a solution on $\mathbb{R}$, then $f_{\mid X}$ is a solution on $X$. However, we have not solved the Cauchy equation on $\mathbb{R}$, only provided a family of solutions: the linear functions. Other solutions exist, whose construction requires the use of the axiom of choice.

We also note that sometimes we can have information about the set of solutions before even having a single one. For example, here, if $f$ and $g$ are two solutions to the Cauchy equation, then $f+g$ is a solution. If $c$ is a real constant, $c f$ is a solution; these last two properties say that the Cauchy equation is linear.

1. This course is not the ideal place to encounter the notion of continuity for the first time, so we will make little use of it. However, here is a definition based on the concept of the limit of a sequence that will suffice here: $f: X \rightarrow \mathbb{R}$ is continuous if for every $x \in X$ and every sequence $x_{n}$ with values in $X$ such that $x_{n} \rightarrow x$, we have that $f\left(x_{n}\right) \rightarrow f(x)$.

## Functional Equations for Affine Functions

Affine functions are ${ }^{2}$ the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following functional equation of "constant slope": there exists a real constant $a$ such that for all $x$ and $y$

$$
\frac{f(x)-f(y)}{x-y}=a
$$

Jensen's Equation Here is Jensen's functional equation

$$
f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}
$$