Uncountably Many Maxima?

Calculus Level 4

Let f:RRf : \mathbb{R} \to \mathbb{R}. Let SS be the set of all strict local maximum of ff. Can SS be uncountable?


  • A strict local maximum is a value xx such that there exists a neighborhood NxN_x of xx where for all yNxy \in N_x, f(y)<f(x)f(y) < f(x). A constant function has no strict local maximum, because f(y)f(x)f(y) \not< f(x); just being equal is not enough.
  • A set is uncountable if there is no injection from it to the natural numbers. In other words, its elements cannot be listed exhaustively in a sequence a1,a2,a3,a_1, a_2, a_3, \ldots.
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