Uncountably Many Maxima?

Calculus Level 4

Let \(f : \mathbb{R} \to \mathbb{R}\). Let \(S\) be the set of all strict local maximum of \(f\). Can \(S\) be uncountable?


  • A strict local maximum is a value \(x\) such that there exists a neighborhood \(N_x\) of \(x\) where for all \(y \in N_x\), \(f(y) < f(x)\). A constant function has no strict local maximum, because \(f(y) \not< f(x)\); 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 \(a_1, a_2, a_3, \ldots\).
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