Continuity of Real Scalar Fields

Continuity

In the case of real scalar fields, the definition of continuity reduces to the following.

Continuity of Real Scalar Fields

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to \mathbb{R}$ be a real scalar field and let $\mathbf{a}\in \mathcal{D}$.

We say that $f$ is continuous at $\mathbf{a}$ if its limit for $\mathbf{x}\to \mathbf{a}$ is $f(\mathbf{a})$, i.e.

$$\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})=f(\mathbf{a}),$$

or $\mathbf{a}$ is an isolated point of $D$.

We say that $f$ is continuous on $S\subseteq \mathcal{D}$ if it is continuous at every $\mathbf{a}\in S$. If $S=\mathcal{D}$, then we just say that $f$ is continuous.

Theorem: Continuity of Real Scalar Fields

Let $f,g:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be real scalar fields.

If $f$ and $g$ are continuous at $\mathbf{a}\in \mathcal{D}$, then so is their product $fg$.

Furthermore, if $g(\mathbf{x})\ne 0$ for all $\mathbf{x}\in \mathcal{D}$, then the product $f/g$ is also continuous at $\mathbf{a}$.

PROOF

TODO