Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0) = 1$ and $f(2x) - f(x) = x$ for all $x \in \mathbb{R}$. If $\mathop {\lim }\limits_{n \to \infty } \left\{ {f(x) - f\left( {\frac{x}{{{2^n}}}} \right)} \right\} = G(x)$, then $\sum\limits_{r = 1}^{10} G \left( {{r^2}} \right)$ is equal to