Upper and lower densities have the strong Darboux property

Let $\mathcal{P}(\mathbf{N})$ be the power set of $\mathbf{N}$. An upper density (on $\mathbf{N}$) is a non-decreasing and subadditive function $\mu^\ast: $\mathcal{P}$({\bf N})\to\bf R$suchthat$\mu^\ast({\bf N}) = 1$and$\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$forall$X \subseteq \bf N$and$h,k \in {\bf N}^+$,where$k \cdot X + h := {kx + h: x \in X}$. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper P'olya, and upper analytic densities are examples of upper densities. We show that every upper density ${\mu }^{\ast }$ has the strong Darboux property, and so does the associated lower density, where a function $f: \mathcal P({\bf N}) \to \bf R$issaidtohavethestrongDarbouxpropertyif,whenever$X \subseteq Y \subseteq \bf N$and$a \in [f(X),f(Y)]$,thereisaset$A$ such that $X\subseteq A\subseteq Y$ and $f(A)=a$. In fact, we prove the above under the assumption that the monotonicity of ${\mu }^{\ast }$ is relaxed to the weaker condition that ${\mu }^{\ast }(X)\le 1$ for every $X\subseteq \mathbf{N}$.