FUNCTIONS FOR WHICH ALL POINTS ARE LOCAL EXTREMA.

Let X be a connected separable linear order, a connected separable metric space, or a connected, locally connected complete metric space. We show that every continuous function f : X → ℝ with the property that every x E X is a local maximum or minimum of f is in fact constant. We provide an example of a compact connected linear order X and a continuous function f : X → ℝ that is not constant and yet every point of X is a local minimum or maximum of f. [ABSTRACT FROM AUTHOR]