Using the technique of changing the unknown function, find all monotonic functions $f$ on $\mathbb{R}$ such that there exist $g$ and $h$ from $\mathbb{R}$ to $\mathbb{R}$ with

$$
f(x+y)=g(x)+h(y)
$$

2. This can be verified, but it can also be taken as a definition
Let $y=0, g(0)=a$ and $h(0)=b$. We then find the two equations

$$
\begin{aligned}
& g(x)=f(x)-b \\
& h(x)=f(x)-a
\end{aligned}
$$

from which $f(x+y)=f(x)+f(y)-a-b$. We make the change of functions $f_{0}(x)=f(x)-a-b$, which yields the Cauchy equation for $f_{0}$. Conclusion: the three functions are affine, more precisely there exists $c$ such that

$$
\begin{aligned}
g(x) & =c x+a \\
h(x) & =c x+b \\
f(x) & =c x+a+b
\end{aligned}
$$

Remark 160. The Jensen equation is a special case of (VII.7) (what are $g$ and $h$?) and we could deduce Exercise 10 from the previous exercise.

## Even Functions, Odd Functions

Definition 161. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be even (resp. odd) if for all $x \in \mathbb{R}$ we have $f(x)=f(-x)$ (resp. $f(x)=-f(-x)$).

The graph of an even function has axial symmetry (with respect to the y-axis); that of an odd function has central symmetry with respect to the origin.

Example 162. The polynomial function $x \mapsto x^{n}$ is even (resp. odd) if and only if $n$ is even (resp. odd).