Smoothness of Real Vector Functions

Definition: Smoothness of a Real Vector Function

A real vector function $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ is smooth iff it is $k$-times continuously partially differentiable for every $k\in \mathbb{N}$.

NOTATION

$$f\in C^{\infty }$$

Definition: Piecewise Smoothness

A real vector function $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ is piecewise smooth if $\mathcal{D}$ can be represented as a disjoint union $\mathcal{D}={\mathcal{D}}_1\cup \cdots \cup {\mathcal{D}}_k$ of finitely many subsets ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_k$ of ${\mathbb{R}}^m$ such that the restrictions $f_{{\mathcal{D}}_1},\dots ,f_{{\mathcal{D}}_k}$ are smooth.

INTUITION

Intuitively, this means that $D$ can be partitioned into finitely many parts such that $f$ is smooth on each part and is not smooth only at a finite number of points in $D$.