Math — Cauchy's functional equations: four rules that pin down a function

These four equations are classics from math competitions and calculus: Cauchy's functional equations and their variants. The core idea is simple: a function's operational property determines its specific form.

Here is each rule, what it means intuitively, and where it shows up.

1. Midpoint rule

f((x+y)/2) = (f(x) + f(y)) / 2  ⟹  f(x) = ax + b

Intuition: when the input takes the midpoint, the output takes the midpoint too. The graph neither bends up nor bends down — it is a perfectly straight line.

Geometry: the midpoint of any two points on the graph still lies on the graph. That is a property unique to linear functions (ax + b).

Where it appears: problems involving midpoints, averages, or smooth linear relationships.

2. Additive rule (Cauchy's original)

f(x + y) = f(x) + f(y)  ⟹  f(x) = ax

Intuition: inputs add, so outputs add. This is the most basic Cauchy functional equation.

Key difference from rule 1: there is no constant term b here. Set x = 0, y = 0 and you get f(0) = f(0) + f(0), so f(0) = 0. The line must pass through the origin.

Real-world example: direct proportionality — "x kg of apples costs f(x), y kg costs f(y), so x + y kg together costs exactly f(x) + f(y)".

3. Logarithmic rule

f(xy) = f(x) + f(y)  ⟹  f(x) = a·ln(x)

Intuition: inputs multiply, outputs add.

The underlying property: this is the defining identity of logarithms, e.g. ln(xy) = ln(x) + ln(y). A logarithm flattens multiplication into addition.

Real-world examples: compound-interest years, decibels in acoustics, information entropy.

4. Exponential rule

f(x + y) = f(x)·f(y)  ⟹  f(x) = c^x

Intuition: inputs add, outputs multiply.

The underlying property: this is the defining identity of exponentials, e.g. c^(x+y) = c^x · c^y. Adding a fixed amount to the input doubles or scales the output.

Real-world examples: cell division, radioactive decay, bacterial growth, population models.

A note on notation: the result is written c^x (equivalently e^{kx} or B·a^x) rather than reusing a from the rules above. There, a was the slope of a line (f(x) = ax, f(x) = a·ln(x)); here c is a base sitting in the exponent position. Different roles, different letters — keeps things unambiguous.

Cheat sheet

Next time a calculus or function problem hands you a functional equation, check it against these four patterns first — matching one of them tells you the function model immediately, and the problem usually cracks open from there.