Exercise 93Calculus › Single-variable calculus › Continuity › Functional equationsL1Added 4 Nov 2026

Additive and multiplicative functions

#Cauchy equation#squares are nonnegative#monotonicity

Statement

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real $x,y$ :
$f(x+y)=f(x)+f(y)\qquad\text{and}\qquad f(xy)=f(x)f(y).$

Hint

Compute $f(x^2)$ in two ways to prove that $f$ takes nonnegative values on $[0,\infty)$ ; deduce monotonicity and use Cauchy's equation.

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Nonnegativity. For $x\ge0$ write $x=t^2$ . Then $f(x)=f(t\cdot t)=f(t)^2\ge0$ .

Monotonicity. If $x\le y$ then $f(y)-f(x)=f(y-x)\ge0$ (additivity and nonnegativity on $[0,\infty)$ ). So $f$ is non-decreasing.

Cauchy. An additive monotone function is linear: $f(x)=cx$ (see the problem "Cauchy's equation with monotonicity").

The constant. Multiplicativity: $f(1\cdot1)=f(1)^2$ , so $c=c^2$ , hence $c\in\{0,1\}$ .

Answer. $f(x)=0$ for all $x$ , or $f(x)=x$ for all $x$ . Both satisfy both equations.

Lesson. The relation $f(t^2)=f(t)^2$ is the standard way to get *positivity*, and positivity gives monotonicity.