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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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We call it odd function.

P.S. We call a function which satisfies $f(-x)=f(x)$ even function.

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  • $\begingroup$ The oddest of all functions are those that are both even and odd. $\endgroup$ Commented Jan 1, 2014 at 16:05
  • $\begingroup$ Thanks for pointing it out. $\endgroup$
    – mathlove
    Commented Jan 1, 2014 at 16:06
  • $\begingroup$ Why the Japanese wikipedia? :) $\endgroup$ Commented Jan 1, 2014 at 16:07
  • $\begingroup$ Oh, sorry, no meaning! $\endgroup$
    – mathlove
    Commented Jan 1, 2014 at 16:08
  • $\begingroup$ I edited it to English one. $\endgroup$
    – mathlove
    Commented Jan 1, 2014 at 16:09
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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.

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