a Stone-Weierstrass Theorem for Function Spaces that do not Contain the Constant Functions

We prove a Stone-Weierstrass theorem for subspaces of C₀(X, R). If an ultraseparating subspace B of C₀(X, R) has a continuous operating function that is neither odd nor affine in every neighbourhood of 0 then B is uniformly dense in C₀(X, R).