Continuity

Continuity

Suppose $\boldsymbol{f}:X\mapsto \mathbb{R}^m$ is a function where $X\subset\mathbb{R}^n$ and some point $\boldsymbol{a}\in X$. $\boldsymbol{f}$ is continuous at $\boldsymbol{a}$ if and only if $\boldsymbol{a}$ is an isolated point or

$$\lim_{\boldsymbol{x}\to\boldsymbol{a}}\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{f}(\boldsymbol{a})$$

If $\boldsymbol{f}$ is continuous at every point in the domain, then $\boldsymbol{f}$ is a continuous function, otherwise, it’s a discontinuous function.

Properties of Continuity

Let $\boldsymbol{f},\boldsymbol{g}:X\mapsto \mathbb{R}^m$ be functions and both of them are continuous at $\boldsymbol{a}\in X$, then the following properties hold:

1. $\boldsymbol{f}\plusmn\boldsymbol{g}$ is also continuous at $\boldsymbol{a}$
2. For any $c\in\mathbb{R}$，$c\boldsymbol{f}$ is also continuous at $\boldsymbol{a}$
3. If $m=1$, then $\boldsymbol{fg}$ is also continuous at $\boldsymbol{a}$
4. If $m=1$ and $\boldsymbol{g}(\boldsymbol{a})\ne 0$, then $\frac{\boldsymbol{f}}{\boldsymbol{g}}$ is continuous at $\boldsymbol{a}$.

Continuity is transitive. Let $\boldsymbol{f}:X\mapsto \mathbb{R}^m$ and $\boldsymbol{g}:Y\mapsto\mathbb{R}^k$, where $X\subset\mathbb{R}^n, \boldsymbol{f}(X)\subset Y\subset\mathbb{R}^m$, and some point $\boldsymbol{a}$ is a limit point of $X$.

Then, if $\lim_{\boldsymbol{x}\to\boldsymbol{a}}\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{L}$ and $\boldsymbol{g}$ is continuous at $\boldsymbol{L}$, we can conclude that $\lim_{\boldsymbol{x}\to\boldsymbol{a}}(\boldsymbol{g}\circ\boldsymbol{f})(\boldsymbol{x})=\boldsymbol{g}(\boldsymbol{L})$．

Moreover, if $\boldsymbol{f},\boldsymbol{g}$ are both continuous functions, then $\boldsymbol{g}\circ\boldsymbol{f}$ is also a continuous function

Smooth

Let function $f:X\mapsto\mathbb{R}, X\subset \mathbb{R}^n$ is open. Let $k$ be a positive integer. We say $f$ is of class $C^k$ if all its partial derivatives of order at most $k$ exist and are continuous.

We say $f$ is smooth, or of class $C^\infin$, if all its partial derivatives of any order exist and are continuous.

For vector-valued functions $\boldsymbol{f}$, the same definitions apply if every component function of $\boldsymbol{f}$ is of class $C^k$ or $C^\infin$ respectively.