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Proposition: Functional Equation of the Exponential Function of General Base (Revised)

Let \(f:\mathbb R\to \mathbb R\) be any continuous real function that obeys the functional equation

\[f(x+y)=f(x)\cdot f(y)\]

for all \(x,y\in\mathbb R\). Then there are two possible cases:

• either \(f\) is constant and equals \(0\), formally \(f(x)=0\) for all \(x\in\mathbb R\), or
• \(f\) equals the exponential function of general base \(f(x)=\exp_a(x)\) for all \(x\in\ \mathbb R\) for the base \(a:=f(1) > 0 \).

Table of Contents

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Proofs: 1

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983