Algebraic Continuity Theorem

Let $f,g:A\to \mathbb{R}$ where $A\subseteq R$ and $c\in A$. Suppose $f,g$ each are continuous at $c$. then:

1. $\alpha f$ (for all $\alpha \in \mathbb{R}$)
2. $f+g$
3. $fg$
   Are continuous at $c$. If further $g(c)\ne 0$ then $\frac{f}{g}$ is continuous at $c$ too.