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: Derivative of an Odd Function

The derivative of an odd function is even.

PROOF

TODO

Theorem: Derivative of an Even Function

The derivative of an even function is odd.

PROOF

TODO