The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman classes to constructing useful counterexamples for some advanced questions. Solutions of this equation are often called additive functions.

Also a few other equations related to this equation are often studied. (Equations which can be easily transformed to Cauchy functional equation or can be solved by using similar methods.)

Is there some overview of basic facts about Cauchy equation and related functional equations - preferably available online?

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    This is also done to some extent to make finding duplicate questions easier, since questions of this type appear at MSE quite frequently. See this meta post for the format of the question (and answer) I have used here. – Martin Sleziak Jun 18 '13 at 10:44
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    Wikihammered per OP request. Since this is a sort of an abstract duplicate question I also added the faq tag. – Willie Wong Jun 18 '13 at 11:39

I have no doubt that many resources about Cauchy functional equations and its relatives are available. But many properties of them have been shown in MSE posts, I will try to provide here list of links to the questions I am aware of. I have made this post CW, feel free to add more links and improve this answer in any way. Several links have been collected also in this post: Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$ $\newcommand{\R}{\mathbb{R}}\newcommand{\Q}{\mathbb{Q}}\newcommand{\Zobr}[3]{{#1}\colon{#2}\to{#3}}$

Cauchy functional equation

We are interested in functions $\Zobr f{\R}{\R}$ such that $$f(x+y)=f(x)+f(y) \tag{C}$$ holds for each $x,y\in\R$.

  • If $f$ is a solution of (C) and $q$ is a rational number, then $f(qx)=qf(x)$ holds for each $x\in\R$.

See e.g. If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$

  • Every continuous solution $\Zobr f{\R}{\R}$ of (C) has the form $f(x)=cx$ for some constant $c\in\R$.

See e.g. this question: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive. or I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$

  • If $\Zobr f{\R}{\R}$ is a solution of (C) which is continuous at some point $x_0$, then it is continuous everywhere.

See e.g. Proving that an additive function $f$ is continuous if it is continuous at a single point

  • Every Lebesgue measurable solution $\Zobr f{\R}{\R}$ of (C) has the form $f(x)=cx$ for some constant $c\in\R$.

See e.g. Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable or Measurable Cauchy Function is Continuous or Prove that if a particular function is measurable, then its image is a rect line or Additivity + Measurability $\implies$ Continuity.

  • If a solution $f$ of (C) is bounded on some interval, them $f(x)=cx$ for some $c\in\R$.

One part of this question is about the proof that locally bounded solution of (C) is necessarily continuous: Real Analysis Proofs: Additive Functions

  • Every integrable solution $\Zobr f{\R}{\R}$ of (C) has the form $f(x)=cx$ for some constant $c\in\R$.

See e.g. How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable

  • There exists non-continuous solutions of (C).

See non-continuous function satisfies $f(x+y)=f(x)+f(y)$, Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?, On sort-of-linear functions, Many other solutions of the Cauchy's Functional Equation

NOTE: The proof of this fact needs at least some form of Axiom of Choice - it is not provable in ZF.

For the role of AC in this result, see this MO thread, and theses questions: Cauchy functional equation with non choice, A question concerning on the axiom of choice and Cauchy functional equation, Is there a non-trivial example of a $\mathbb Q-$endomorphism of $\mathbb R$?, What is $\operatorname{Aut}(\mathbb{R},+)$?

  • The graph $G(f)=\{(x,f(x)); x\in\R\}$ is a dense subset of $\R^2$ for every non-continuous solution of (C).

See e.g. Graph of discontinuous linear function is dense

  • If we have the equation (C) for function $\Zobr f{[0,\infty)}{[0,\infty)}$, then every solution is of the form $f(x)=cx$.

See If $f:[0,\infty)\to [0,\infty)$ and $f(x+y)=f(x)+f(y)$ then prove that $f(x)=ax$

Related equations

There are several functional equations that are closely related to Cauchy equation. They can be reduced to it by some methods or solved by very similar methods as the Cauchy equation.

The equation $f(x+y)=f(x)f(y)$

$$f(x+y)=f(x)f(y) \tag{1}$$

  • If $f$ is a solution of (1), then either $f=0$ (i.e., $f$ is constant function equal to zero) or $f(x)>0$ for each $x\in\R$.

See e.g. Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$? and A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere

  • If $f$ is a solution of (1), $x\in\Q$ and if we denote $a=f(1)$, then $f(x)=a^x$.

  • Every continuous solution of (1) has the form $f(x)=a^x$ for some $a\ge 0$.

See e.g. continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$ or How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?

  • If a solution of (1) is continuous at $0$, then it is continuous everywhere.

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

  • There are non-continuous solutions of (1).

They can be obtained from non-continuous solutions of (C).

See Is the group isomorphism $\exp(\alpha x)$ from the group $(\mathbb{R},+)$ to $(\mathbb{R}_{>0},\times)$ unique?

  • If a solution of (1) is differentiable at $0$, then it is differentiable everywhere and $f'(x)=f(x)f'(0)$.

See Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists and Differentiability of $f(x+y) = f(x)f(y)$

The equation $f(xy)=f(x)+f(y)$

$$f(xy)=f(x)+f(y) \tag{2}$$

Examples of functions where $f(ab)=f(a)+f(b)$

The equation $f(xy)=f(x)f(y)$

$$f(xy)=f(x)f(y) \tag{3}$$

  • Every continuous solution of (3) has the form $f(x)=x^a$

See If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t

  • A solution of (3) which is continuous at $1$ is continuous for each $x>0$.

See Functional equation $f(xy)=f(x)+f(y)$ and continuity

  • If a function fulfills both (C) and (3), then either $f(x)=0$ or $f(x)=x$ (i.e., it is either zero function or the identity).

Notice that we do not require continuity here. See Functional equations $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$

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