Differentiability

A map $𝑓 : 𝐷 \subseteq ℝ \to 𝑅 \subseteq ℝ$ is $𝑛$ -differentiable at a point $𝑥 0 \in 𝑈$ iff it has an $𝑛$ -th derivative at that point, and thus all derivatives up to $𝑛$ . #m/def/anal Moreover $𝑓$ is called $𝑛$ -differentiable if it is differentiable at every $𝑥 0 \in 𝑈$ . $𝐶 𝑛$ is the set of all $𝑛$ -differentiable functions with a continuous $𝑛$ th derivative, and is called a differentiability class, and $𝑛 \leq 𝑚 ⟹ 𝐶 𝑛 \subseteq 𝐶 𝑚$ . In particular,

$𝐶 0$ is the class of all continuous functions;
$𝐶 𝜔$ of analytic functions; and
$𝐶 \infty$ of infinitely differentiable functions¹.

Generalizations

Complex functions

In complex analysis all differentiable functions are analytic and infinitely differentiable. Such a function is called holomorphic.

Open subsets of real coördinate space

Differentiability generalizes naturally to higher dimensional Real coördinate space (and open subsets thereof). A function $𝑓 : ℝ 𝑁 \to ℝ 𝑀$ is $𝐶 𝑛$ iff it has all $𝑛$ -th order partial derivatives.

Arbitrary subsets of real coördinate space

Let $𝑋 \subseteq ℝ 𝑁$ be inhabited. A function $𝑓 : 𝑋 \to ℝ 𝑀$ is $𝐶 𝑘$ iff every $𝑥 \in 𝑋$ has an open neighbourhood $𝑈 \subseteq ℝ 𝑁$ with a $𝐶 𝑘$ extension $𝐹 : 𝑈 \to ℝ 𝑀$ such that $𝐹 (𝑦) = 𝑓 (𝑦)$ for all $𝑦 \in 𝑋 \cap 𝑈$ . #m/def/geo/diff

By considering real submanifolds, this yields the notion of differentiability for maps between such manifolds.

Map between manifolds

Let $𝑓 : 𝑀 \to 𝑁$ be a map between manifolds of dimension $𝑚$ and $𝑛$ respectively. The $𝐶 𝑘$ class is only well defined if $𝑀$ and $𝑁$ are $𝐶 𝑘$ differentiable manifolds. Let $𝑝 \in 𝑀$ . $𝑓$ is called $𝑘$ -differentiable at $𝑝 \in 𝑀$ iff there exists a chart $(𝑈, 𝜑)$ containing $𝑝$ and $(𝑉, 𝜓)$ containing $𝑓 (𝑝)$ such that $𝜓 𝑓 𝜑 - 1$ is $𝑘$ -differentiable at $𝑝$ . #m/def/geo/diff $𝑓$ is called $𝐶 𝑘$ iff it is $𝑘$ -differentiable everywhere.

#state/tidy | #lang/en | #SemBr

Often called smooth, however the exact meaning of this term varies between authors, e.g. sometimes $𝐶 1$ functions are referred to as smooth. ↩