Cauchy's Functional Equation

This is the most famous functional equation and a cornerstone of the topic.

The Equation

Solution Steps

1. Zero: $f(0+0)=f(0)+f(0)⟹f(0)=0$.
2. Integers:

• $f(1)=c$.

• $f(2)=f(1+1)=f(1)+f(1)=2c$.

• By induction, $f(n)=cn$ for $n\in \mathbb{N}$.

3. Negatives: $f(x-x)=f(x)+f(-x)⟹0=cx+f(-x)⟹f(-x)=-cx$.

1. Rationals: Let $x=p/q$.

• $f(q\cdot \frac{p}{q})=qf(\frac{p}{q})$ (by additivity).

• $f(p)=qf(\frac{p}{q})$.

• $cp=qf(\frac{p}{q})⟹f(\frac{p}{q})=c\frac{p}{q}$.

Conclusion: Over $\mathbb{Q}$, the only solutions are $f(x)=cx$.

Extension to Real Numbers

Over $\mathbb{R}$, $f(x)=cx$ is the unique solution only if we add a regularity condition:

1. $f$ is continuous.
2. OR $f$ is monotonic.
3. OR $f$ is bounded on some interval.

Without these, "wild" non-linear solutions exist (using Hamel Bases).

Related Equations

1. Jensen's FE: $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}⟹f(x)=ax+b$.
2. Logarithmic: $f(xy)=f(x)+f(y)⟹f(x)=c\ln x$.
3. Exponential: $f(x+y)=f(x)f(y)⟹f(x)=a^x$.

Practice Problems