Let \(f \colon \mathbb{R} \to \mathbb{R} \) be a continuous function such that \[f(x+1)=\frac{1}{2}{f(x)}\space \forall x \in \mathbb{R},\] and let \(a_n=\int_{0}^{n} {f(x)}{dx}\) for all integers \(n\geq {1}.\) Then \[\lim_{n\to \infty} {a_n}\space \text{exists and equals}\space 2\int _{0}^{1} {f(x)}{dx}.\]