Entire functions agreeing with $e^x$ on $\RR$

Find all entire functions $f$ such that $f(x) = e^x$ on $\RR$.

Solution

The function $g(z) \da f(z) - e^z$ is entire and identically zero on $\RR$, which contains a limit point. So $g(z) \equiv 0$ on $\CC$, meaning $f(z) = e^z$ is the only such function.