A function $f\colon \mathcal{P}(\mathbf{N})\to [0,1]$ is said to have the *Darboux property* whenever for all $X \subseteq \mathbf{N}$ and $y \in [0,f(X)]$, there exists $Y \subseteq X$ such that $f(Y)=y$.

Moreover, $f$ is said to be *monotone* if $f(X)\le f(Y)$ whenever $X\subseteq Y$ and *subadditive* if $f(X\cup Y) \le f(X)+f(Y)$ for all $X,Y \subseteq \mathbf{N}$.

Question.Let $f\colon \mathcal{P}(\mathbf{N})\to [0,1]$ be a monotone subadditive function with the Darboux property such that $f(\mathbf{N})=1$. Does there exist necessarily afinitepartition $\{A_1,\ldots,A_k\}$ of $\mathbf{N}$ for which $$ f(A_1), \ldots, f(A_k) \in (0,1)? $$

A simpler version of this question has been posted on MSE one week ago here, showing that in general the answer is negative if $k$ is bounded.

The question comes from the study of properties of a class of functions satisfying certain axioms, which can be called as "upper densities" (with this respect, you can see this MO question). Here, instead, there is MO question related to the definition of Darboux property.

A related construction (which does not ask for subadditivity) by Salvo Tringali has been provided here.