Definition:Odd Function

Definition

Let $X \subset \R$ be a symmetric set of real numbers:

$\forall x \in X: -x \in X$

A real function $f: X \to \R$ is an odd function if and only if:

$\forall x \in X: \map f {-x} = -\map f x$

Examples

Identity Function

Let $I_\R: \R \to \R$ denote the identity function on $\R$.

Then $I_\R$ is an odd function.

Cube Function

Let $f: \R \to \R$ denote the cube function on $\R$.

$\forall x \in \R: \map f x = x^3$

Then $f$ is an odd function.

Also see

• Definition:Even Function

• Results about odd functions can be found here.

Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 4$. Even and Odd Functions: $(2)$
• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(h)}$ Even and Odd Functions
• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $1$. Functions: $1.5$ Trigonometric or Circular Functions: $1.5.2$ Sine Function
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): odd function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): odd function
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): odd function