Let \(f:\mathrm{ℝ}\to \mathrm{ℝ}\)be a continuous function satisfying\(f(0)=1\)and \(f(2\mathrm{x})-f(\mathrm{x})=\mathr…

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

[JEE Main 2025, 4 Apr (Shift 1)]