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