Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Show that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}). \qquad(1)$$ Where $\{x\}$ is the fractional part of $x$.

It's easy to prove that $$\{x\}=\frac{1}{2}+\int_0^1 (y-\mathbf{1}_{[\{x\},1]}(y))\,\mathrm{d}y,\qquad (2)$$ Where $\mathbf{1}_A(x)=\begin{cases} 1, & x\in A, \\ 0, & x\notin A. \end{cases}$

How to prove (1)? Is the (2) useful?

I find this problem in G.Tenenbaum's book in the page 118.

Gérald Tenenbaum. Introduction to Analytic and Probabilistic Number Theory, Cambridge: Cambridge University Press, 1995.