Continuity properties

If $f(x)$ and $g(x)$ are continuous at $x=a$, and if $c$ is a constant, then $f(x)+g(x)$ is continuous at $x=a$, $f(x)-g(x)$ is continuous at $x=a$, $cf(x)$ is continuous at $x=a$, $f(x)g(x)$ is continuous at $x=a$, and $\frac{f(x)}{g(x)}$ is continuous at $x=a$ as long as $g(a) \ne 0$.

These rules follow directly from the limit laws. For instance, to see that $f(x)+g(x)$ is continuous at $x=a$, we need to show that $\displaystyle{\lim_{x \to a} (f(x)+g(x))}=f(a)+g(a)$. But

Our next property involves composite functions:

Theorem: If $f(x)$ is continuous at $x=b$, and if $\displaystyle{\lim_{x \to a}} g(x) = b$, then $\displaystyle{\lim_{x \to a} f(g(x)) = f(b)}$.

To see this, suppose that $x$ is close to (but not equal) to $a$. Then $g(x)$ is close to $b$, since $\displaystyle{\lim_{x \to a}g(x)=b}$. Let $y=g(x)$. Since $f$ is continuous at $b$, whenever $y$ is close to $b$, $f(y)$ is close to $f(b)$. But that makes $f(g(x))$ close to $f(b)$.

The following video goes over these properties and how to use them.