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September 15

Continuous function that is "even" at some points but "odd" at others

Let $D$ be either the real numbers ( $\mathbb {R}$ ) or the nonzero real numbers ( $\mathbb {R} \setminus \{0\}$ ). Then, does there exist a continuous function $f:D\to \mathbb {R}$ that is neither even nor odd but such that $\forall x\in D\,(f(-x)\in \{f(x),-f(x)\})$ ? Note that since $\mathbb {R}$ is a connected Hausdorff space, any such function with domain $\mathbb {R}$ must have at least one x-intercept. GeoffreyT2000 (talk) 02:19, 15 September 2019 (UTC)[reply]

Yes, and you have given yourself a hint for construction. Try to create a function which piecewise alternates being even and odd, changing over at its x-intercepts. -- ToE 04:17, 15 September 2019 (UTC)[reply]