Suppose that $f:\mathbb{R}\to\mathbb{R}$ is continuous, and consider the linear interpolation
$$p(x)=\frac{b-x}{b-a}f(a)+\frac{x-a}{b-a}f(b)$$ where $x\in [a,b]\subset{R}$. Suppose that $b-a=:h\downarrow 0$, is there a way to derive some error estimation in terms of Landau-$O$-notation such as $$f(x)=p(x)+O(h^2)\qquad h\downarrow 0$$ since this is somehow suggested in a paper with that I have to work with. I don't see a clear way since there is no possibility to use error bounds via Taylor expansion.