Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f(0) = 1 \) and \( f(2x) - f(x) = x \) for all \( x \in \mathbb{R} \). If \( \displaystyle \lim_{n \to \infty} \Big[ f(x) - f\!\left(\dfrac{x}{2^n}\right) \Big] = G(x) \), then \( \displaystyle \sum_{r=1}^{10} G(r^2) \) is equal to:
