I wish to show that a function which is "essentially constant" (defined shortly) can't be a good classifier (machine learning). For this i need to estimate the "complexity" of such a class of functions.

So let $\mathcal X$ be an abstract set (we may assume has metric structure, e.g $\mathbb R^d$). Given $0 < \alpha \ll 1$, define

$$ \mathcal H_\alpha := \{h: \mathcal X \rightarrow [0, 1]\text{ s.t } | \exists \bar{h} \in \mathbb [0, 1] \text{ veryfing } |h(x) - \bar{h}| \le \alpha\;\forall x \in \mathcal X\}. $$

# Questions

**(A)** What is the VC dimension or fat-shattering dimension of $\mathcal H_\alpha$ ? Good lower and upper bounds thereof would be just as important.

**(B)** I'd also be interested in metric complexity measures for $\mathcal H_\alpha$ (covering number, metric entropy, etc.).
Good lower and upper bounds thereof would be just as important.

# Observations

Theorem 2 of this 1997 paper shows that there exists absolute constants $c_1,c_2$ such that for any $\mathcal H \subseteq [0,1]^{\mathcal X}$, the following bounds metric entropy and fat-shattering dimension holds

$$ \operatorname{fat}_{\mathcal {H}}(4\epsilon)/32 \le \max_{P}\log_2(\mathcal N(\epsilon,\mathcal H,\mathcal L_1(dP))) \le c_1 \operatorname{fat}_{\mathcal H}(c_2 \epsilon)(\log_2(1/\epsilon))^2. $$ So if we can estimate $\operatorname{fat}_{\mathcal H_\alpha}(\gamma)$, then we're done!

**Answer to one-dimensional case.**
A user has given an explicit computation of $\operatorname{fat}_{\mathcal H_\alpha}(\gamma)$ in the simple one-dimensional case $\mathcal X = [0, 1]$.