Upper and lower densities have the strong Darboux property

Let P(N) be the power set of N. An upper density (on N) is a nondecreasing and subadditive function μ⋆:P(N)→R such that μ⋆(N)=1 and μ⋆(k⋅X+h)=1kμ⋆(X) for all X⊆N and h,k∈N+, where k⋅X+h:={kx+h:x∈X}. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper Pólya, and upper analytic densities are some examples of upper densities. We show that every upper density μ⋆ has the strong Darboux property, and so does the associated lower density, where a function f:P(N)→R is said to have the strong Darboux property if, whenever X⊆Y⊆N and a∈[f(X),f(Y)], there is a set A such that X⊆A⊆Y and f(A)=a. In fact, we prove the above under the assumption that the monotonicity of μ⋆ is relaxed to the weaker condition that μ⋆(X)≤1 for every X⊆N.