Smooth functions on Euclidean space

Definition

Let $k$ be a nonnegative integer. $U\subseteq {\mathbb{R}}^n$ is an open set. A real-valued function $f:U\to \mathbb{R}$ is said to be ${\mathcal{C}}^k$ at $\mathbf{p}\in U$ if its partial derivatives

$$\frac{{\partial }^{j}f}{\partial x^{i_1}\cdots \partial x^{i_j}}(\mathbf{p})$$

of all orders $j\le k$ exist and are continuous at $\mathbf{p}$. The function $f:U\to \mathbb{R}$ is ${\mathcal{C}}^{\infty }$ at $\mathbf{p}$ if it is ${\mathcal{C}}^k$ for all $k\ge 0$;

A vector-valued function $f:U\to {\mathbb{R}}^m$ is said to be ${\mathcal{C}}^k$ at $\mathbf{p}$ if all of its component functions $f^1,\dots ,f^m$ are ${\mathcal{C}}^k$ at $\mathbf{p}$.

We say that $f:U\to {\mathbb{R}}^m$ is ${\mathcal{C}}^k$ on $U$ if it is ${\mathcal{C}}^k$ at every point in $U$. A similar definition holds for a ${\mathcal{C}}^{\infty }$ function on an open set $U$. We treat the terms ${\mathcal{C}}^{\infty }$ and smooth as synonymous.

A neighborhood of a point in ${\mathbb{R}}^n$ is an open set containing the point. The function $f$ is real-analytic at $\mathbf{p}$ if in some neighborhood of $\mathbf{p}$ it is equal to its Taylor series at $\mathbf{p}$:

$$\begin{array}{rl}f(\mathbf{x})=f(\mathbf{p})& +\sum _i\frac{\partial f}{\partial x^i}(\mathbf{p})(x^i-p^i)\\ & +\frac{1}{2!}\sum _{i,j}\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(\mathbf{p})(x^i-p^i)(x^j-p^j)\\ & +\cdots +\frac{1}{k!}\sum _{i_1,\dots ,i_k}\frac{{\partial }^{k}f}{\partial x^{i_1}\cdots \partial x^{i_k}}(\mathbf{p})(x^{i_1}-p^{i_1})\cdots (x^{i_k}-p^{i_k})+\cdots \end{array}$$

i.e. If $S_k$ represents symmetric group of $[k]$, then:

$$f(\mathbf{x})=\sum _{k=0}^{\infty }\sum _{i_1{i}_2\cdots i_k\in S_k}\frac{1}{k!}\frac{{\partial }^{k}f}{\partial x^{i_1}\cdots \partial x^{i_k}}(\mathbf{p})(x^{i_1}-p^{i_1})\cdots (x^{i_k}-p^{i_k})$$

Properties

1. A real-analytic function is necessarily $C^{\infty }$ , because a convergent power series can be differentiated term by term in its region of convergence