= Extreme points of a real continuous-function unit ball
{title2=$|f(x)|=1\quad(x\in K)$}
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.
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