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Local Extrema and Nonopenness Points of Continuous Functions

Abstract We give a short argument showing that the set of openness points of a continuous function from a metric space X into a metric space Y is of type Gδ. If X is locally connected and Y:= ℝ, the set of nonopenness points (of type Fσ) coincides with the set of points of extrema of f. We discuss which Fσ sets can be equal to the set of points of extrema for a continuous function f from ℝ into ℝ, and we present a short survey of the known results on this topic.