Are all useful models true within +-eps?

I like the phrase “All models are wrong, but some are useful”, attributed to George E. P. Box. It encourages humility and pragmatism, encourages us to watch out for when our models aren’t useful, to not be too attached to them.

However, I also wonder if this is true: Are all models that are “useful” true within some finite epsilon on some non-null domain? Say that you can observe the inputs to a system $x \in R^n$ and want to predict output $y = f(x) \in R^m$. Are there any models $\hat{f}(x)$ that are “useful” for which we don’t have a non-null $B \sube R^n$ and $0 < \epsilon < \infin$ such that $\forall x \in B \textrm{,} \abs{\hat{f}(x) – f(x)} \le \epsilon$ and also $\exist x_1, x_2 \in B, \abs{\hat{f}(x_1) – \hat{f}(x_2)} > \epsilon$. The second condition is supposed to rule out choosing an $\epsilon$ so large that the first condition (truthiness) is satisfied trivially.

At least, this seems like a useful thing to consider for any model: What are the limits within which we expect it to hold exactly? I guess in a more sophisticated version of these proposed conditions, you could allow the epsilon to vary with the input value. E.g., if you’re trying to predict the position of a particle and your input is time, you may need to allow larger bounds as time increases. In fact, maybe this is a counterexample — if variability increases with $x$, then for any interval you select, selecting $\epsilon$ large enough to work for the largest $x$ will make it impossible to satisfy the second condition.