Sum of real numbers equal to integer number

Question

Let $t_1, t_2, ..., t_n \in \mathbb{N}_{>0}$, $t = \sum_{i=1}^n t_i$, and $s \in \{n,n+1, ..., t\}$. Does an integer function $f: \mathbb{R}_{>0} \to \mathbb{N}_{>0}$ such that $ \sum_{i=1}^{n}f\left(s \cdot\dfrac{t_i}{t}\right) = s$ exist?