Real Functions

Parity

Definition: Even Function

A real function $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ is even iff

$$f(-x)=f(x)\forall x\in \mathcal{D}$$

Definition: Odd Function

A real function $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ is odd if

$$f(-x)=-f(x)\forall x\in \mathcal{D}$$

Properties

Theorem: Derivatives of Odd and Even Functions

If an odd function is differentiable, then its derivative is an even function.

If an even function is differentiable, then its derivative is an odd function.

PROOF

TODO

Theorem: Operations with Odd and Even Functions

Sums:

• The sum of two even functions is an even function.
• The sum of two odd functions is an odd function.
• The sum of an even function and an odd function is neither even nor odd, unless one of the functions is always zero.

Differences:

• The difference between two even functions is an even function.
• The difference between two odd functions is an odd function.

Products and quotients:

• The product and quotients of two even functions are even functions.
• The product and quotients of two odd functions are even functions.
• The product and quotients of an even function and an odd function are odd functions.

Compositions:

• The Composition $f\circ g$ of any function $g$ with an even function $f$ is an even function.
• The Composition $f\circ g$ of an even function $g$ with an odd function $g$ is an even function.
• The Composition of two odd functions is an odd function.

Inverse:

• The inverse of an odd injection is an odd function.

PROOF

TODO