Smooth Multivariate Functions

Why

What is the natural generalization of a smooth function to functions defined on sets of $\R ^k$.¹

Definition

Let $U \subset \R ^d$ be an open set (see Real Open Sets). A function $f: U \to \R $ is smooth if all its partial derivatives exists and are continuous.

More generally, let $X \subset \R ^d$. A function $f: X \to R$ is smooth if there exists an open set $U \subset \R ^d$ and a smooth $F: U \to \R $ so that $F(x) = f(x)$ for all $x \in U \cap X$.

Example

The identity map is smooth. In other words, let $f: \R ^d \to \R $ be so that $X \subset \R ^d$. Then $f: X \to \R $ s

Properties

1. Future editions will expand. ↩︎