# What do you call a function with the property $f(-x)=-f(x)$?

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?

• en.wikipedia.org/wiki/Even_and_odd_functions Commented Jan 1, 2014 at 16:00
• Odd function. Google it. Commented Jan 1, 2014 at 16:02
• I posted an answer which clarify the definition.
– user63181
Commented Jan 1, 2014 at 17:11

We call it odd function.

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

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

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.