We denote by $C[0, 1]$ the space of continuous functions on $[0, 1]$ under the supremum norm, equipped with the Borel sigma algebra.

A covering of $C[0, 1]$ is a (possibly countably infinite) collection of sets $E_i$ such that $\cup_i E_i = C[0, 1]$.

A covering is said to have finite eccentricity if there exist constants $0 < c \leq C$ such that each $E_i$ contains a open ball of radius $c$ and is contained within an open ball of radius $C$. We call $c$ and $C$ the eccentricity constants of the cover.

A covering is said to be locally finite if every $f \in C[0, 1]$ lies in finitely many of the $E_i$.

Question: Let $\epsilon > 0$ and $0< \delta < 1$ be arbitrary. Does there exist a locally finite covering of $C[0, 1]$ of finite eccentricity with eccentricity constants $(1-\delta)\epsilon$ and $(1+\delta)\epsilon$?