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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