I stumbled over this approximization for a flow-map $m:\Omega\rightarrow \mathbb{R}^2, \Omega \subset \mathbb{R}^2$ in this paper, pages 4-5. The authors approximate the map by integrating it over subsets of its domain. I dont understand, why we can do such a thing to approximate a function.
For example let $m(x)=x^2$ and for $\Omega=[0,1]$ let the discretization given by $\{0,h,2h,...,1\}, 1 = Nh, N\in \mathbb{N}$. If I apply the technique from above, I get that the approximation error can never vanish by just changing the grid. $$ |m(x_i) -\tilde{m}(x_i)| = |x_i^2 - \int_{x_i}^{x_i+\Delta x}x^2dx| \\ = |x_i^2 - \frac{1}{3}((x_i+\Delta x)^3 -x_i^3)|\overset{\Delta x \rightarrow 0}{\rightarrow} |x_i^2| $$
Is there something, that I do not see here?