Continuity of Real Scalar Fields

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 ${\mathbf{x}}_0\in \mathcal{D}$, then so is their product $fg$.

Furthermore, if $g(\mathbf{x})\ne 0$ for all $\mathbf{x}\in \mathcal{D}$, then their quotient $f/g$ is also continuous at ${\mathbf{x}}_0$.

PROOF

TODO