Wikipedia:Reference desk/Archives/Mathematics/2019 September 15

= 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)
 * 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)