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

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

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