For a uniform grid
$$x_n = -1 + nh$$
where $h = \frac{2}{N}$ and the integration rule
$$I_N(f) = h\sum_{n=0}^{N-1}f(x_n)$$
which corresponds to a left hand Riemann sum or to integrating an interpolant in $\mathscr{S}_0^{-1}$, how would I show that this is an $\mathcal{O}(h)$ method?
Let $ε=\max_{x,y\in [-1,1]:\;|x-y|\le h}|f(x)-f(y)|$. Then $$ \left|\int_{x_k}^{x_{k+1}} f(s)\,ds-f(x_k)(x_{k+1}-x_k)\right|\le\int_{x_k}^{x_{k+1}} \Bigl|f(s)-f(x_k)\Bigr|\,ds\le ε·(x_{k+1}-x_k) $$ Now if you can assume that $f$ is Lipschitz or continuously differentiable, then $ε=O(h)$ which would give the desired result.
