Algebraic_Continuity_Theorem

Assume $f:A \rightarrow \mathbb{R}$ and $g:A \rightarrow \mathbb{R}$ are continuous at a point $c \in A$. Then,

1. $kf\left. (x) \right.$ is continuous at $c$ for all $k \in \mathbb{R}$.

2. $f\left. (x) \right. + g\left. (x) \right.$ is continuous at $c$.

3. $f\left. (x) \right.g\left. (x) \right.$ is continuous at $c$.

4. $f\left. (x) \right./g\left. (x) \right.$ is continuous at $c$, provided the quotient is defined.

Proof: