Locally constant functions

Let X be a compact Hausdorff space and M a metric space. ${\displaystyle E_0(X,M)}$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which ${\displaystyle E_0(X,M)}$ is all of C(X,M). These include βℕ\ℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of ${\displaystyle E_0(X,M)}$ as a normed linear space. We also build three first countable Eberlein compact spaces, F,G,H, with various ${\displaystyle E_0}$ properties. For all metric M, ${\displaystyle E_0(F,M)}$ contains only the constant functions, and ${\displaystyle E_0(G,M)=C(G,M)}$. If M is the Hilbert cube or any infinite-dimensional Banach space, then ${\displaystyle E_0(H,M)\ne C(H,M)}$, but ${\displaystyle E_0(H,M)=C(H,M)}$ whenever ${\displaystyle M\subseteq {ℝ}^n}$ for some finite n.