 Extreme points of a real continuous-function unit ball

For compact Hausdorff $K$, the extreme points of the real unit ball in the space of continuous functions on a compact space are precisely its continuous sign-valued functions. A point with $|f|<1$ permits a nonzero continuous bump perturbation supported where there is uniform slack, while pointwise equality at an endpoint of $[-1,1]$ forces both functions in a midpoint decomposition to agree. For the Cantor set, clopen indicators give the perturbations directly.