A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is an "easy estimator" if any point on $f$, $(x_0,y_0)$, is near a lattice point $([x_0],[y_0])$ then $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$.
In other words:
If for any $(x_0,y_0)$ on $f(x)$ where we have $\sqrt{(y_0-[y_0])^2+(x_0 - [x_0])^2} \leq \delta$ then it is also the case that $f([x_0])=[y_0]$
Example of an "easy" easy estimator:
$f(x)=x$
$f(x)=x$ is an easy esitmator because if $(x,y)$ is close to $([x],[y])$ then $([x],[y])$ satisfies $f(x)=y$.
Questions.
What is the largest possible $\delta$ for $f(x)=x$?
What is the largest possible $\delta$ for any $f$?
How about some more examples? Especially non-linear functions?
Consider thre finite sequences of integers $(p_n)_{1\leq n\leq N}$, $(q_n)_{1\leq n\leq N}$ and $(a_n)_{1\leq n\leq N}$, such that for all $n$ $q_n\neq0$ and, without loss of generality, $p_n\wedge q_n=1$. Then the function $$f(x)=\sum_{n=1}^N \left\lvert \frac{p_n}{q_n} x-a_n\right\rvert\tag{1}$$ satisfies your requirements. The graph of $f$ has an irregular sawtooth shape. It is easy to see that the slope of $f$ takes the values $$\frac pq=\sum_{n=1}^N\epsilon_n\left\lvert\frac{p_n}{q_n}\right\rvert,$$ where $\epsilon_n=0$ or $1$.
The minimal distance between a line of slope $\frac pq$ and a point of integer coordinates that does not belong is the value of $\delta$ for this $f$ : $$\delta=\frac{1}{\sqrt{p^2+q^2}}.\tag{2}$$ So $\delta(f)$ is the maximum of expression (2) for all possible choices of $(\epsilon_n)$.