What is this property called? The domain and codomain of the function can be for example $\mathbb Z^n$, $\mathbb Q^n$ or $\mathbb R^n$ ($n>0$), potentially excluding the $0$ point. Examples: $f(x)=ax^k$ ($k$ being an odd integer), rotation in the plane around the origin by a fixed angle, $f(x)=\operatorname{sgn}(x)g(|x|)$ (where $g$ is some other function), etc.
I'm thinking of calling it "symmetric around $0$", does that sound right?
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Davide Giraudo
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2http://en.wikipedia.org/wiki/Even_and_odd_functions – lab bhattacharjee Jan 01 '14 at 16:00
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2Odd function. Google it. – DonAntonio Jan 01 '14 at 16:02
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I posted an answer which clarify the definition. – Jan 01 '14 at 17:11
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We call it odd function.
P.S. We call a function which satisfies $f(-x)=f(x)$ even function.
mathlove
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The oddest of all functions are those that are both even and odd. – Harald Hanche-Olsen Jan 01 '14 at 16:05
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The article says "let f(x) be a real-valued function of a real variable", is it ok to extend it to the sets I mentioned (or whatever context where "-x" makes sense)? – aditsu quit because SE is EVIL Jan 01 '14 at 16:13
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Oh, you mean the part where it says "This includes additive groups, all rings, all fields, and all vector spaces", ok – aditsu quit because SE is EVIL Jan 01 '14 at 16:21
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Just to clarify the definition:
A function defined on a domain $D$ is called odd function if:
- $-x\in D$ whatever $x\in D$
- $f(-x)=-f(x)\;\;\forall x\in D$
Notice that the first point is very important although it is often omitted. For example the function $$f\colon [0,\pi]\rightarrow\mathbb R,\quad x\mapsto\sin x$$ isn't an odd function since the first point isn't fullfiled.