---
tag: SRAN4EKWS0
kind: concept
name: Functional equation
msc: [39B05, 39B52]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_equation
same_as:
  wikidata: Q594452
  wikipedia: Functional equation
---

## Summary

A functional equation is an equation whose unknown is a [[function|0EPAT51DXQ]] and which must hold for all values of some variables.
Both sides are built by composing the unknown function with known operations.
Classic examples are [[Cauchy's functional equation|ZBMEVWN100]] $f(x + y) = f(x) + f(y)$ and the rule $\Gamma(x + 1) = x\,\Gamma(x)$ of the gamma function.
By convention, equations with derivatives or integrals of the unknown are called differential or integral equations instead.
Functional equations characterize many classical functions, such as the exponential, the logarithm and the power functions, by a few simple rules.

## Explanations

### L1 human

In school algebra the unknown is a number; in a functional equation the unknown is a whole function.
One looks for every function that obeys a rule for all inputs.
For example, the rule $f(x + y) = f(x) f(y)$ says that adding inputs multiplies outputs.
The continuous solutions on the real line are $f = 0$ and $f(x) = a^x$ with $a > 0$, so the rule pins down the exponential functions.